Consider the following example: Assuming for a moment that birthdays are evenly distributed throughout the year, if you're sitting in a room with forty people in it, what are the chances that two of those people have the same birthday? For simplicity's sake, we'll ignore leap years. A reasonable, intelligent person might point out that the odds don't reach 100% until there are 366 people in the room (the number of days in a year + 1)... and forty is about 11% of 366... so such a person might conclude that the odds of two people in forty sharing a birthday are about 11%. In reality, due to Math's convoluted reasoning, the odds are about 90%. This phenomenon is known as the *Birthday Paradox*.

If the set of people is increased to sixty, the odds climb to above 99%. This means that with only sixty people in a room, even though there are 365 possible birthdays, it is almost certain that two people have a birthday on the same day. After making these preposterous assertions, Math then goes on to rationalize its claims by recruiting its bastard offspring: numbers and formulas.

It's tricky to explain the phenomenon in a way that feels intuitive. You can consider the fact that forty people can be paired up in 780 unique ways, and it follows that there would be a good chance that at least one of those pairs would share a birthday. But that doesn't really satisfy the question for me, it just feels marginally less screwy. So I did something quite out of character: I crunched the numbers. The values rapidly become unmanageable, but the trend is clear:

# of people | Possible combinations of birthdays | # of those combinations where at least two birthdays fall on the same day | % of combinations where two people have same birthday |
---|---|---|---|

1 | 365 | 0 | 0.0% |

2 | 133,225 | 365 | 0.2% |

3 | 48,627,125 | 398,945 | 0.8% |

4 | 17,748,900,625 | 290,299,465 | 1.6% |

5 | 6,478,348,728,125 | 175,793,709,365 | 2.7% |

6 | 2,364,597,285,765,625 | 95,677,479,012,025 | 4.0% |

7 | 863,078,009,304,453,125 | 48,535,798,679,910,725 | 5.6% |

8 | 315,023,473,396,125,390,625 | 23,417,361,992,539,211,425 | 7.4% |

9 | `IF YOU PUSH THAT EQUALS BUTTON I WILL MAKE YOU BLEED. SINCERELY, YOUR CALCULATOR. END OF LINE.` |
`5318008` |
--- |

Only calculating up to eight people, we see that of the three hundred fifteen *quintillion* possible combinations of birthdays the group has, 7.4% of cases-- or about one in thirteen-- result in two of them having the same birthday. As each person is added, the odds do not increase linearly, but rather they curve upwards rapidly. This trend continues up to around twenty-three people, where the curve hits 50% odds, and the rate of increase starts going down. It practically flattens out when fifty-seven people are considered, and the odds rest at about 99%. Though it may not be intuitive, the numbers follow the pattern quite faithfully.

So does this mean that you can walk into a math class of forty students, bet them that at least two people in the room share a birthday, and win 90% of the time? Not exactly. In real life, where Math is not particularly welcome, birthdays are not distributed perfectly throughout the year. More people are born in the springtime, which throws the numbers off. Also, as a result of the way that hospitals operate, more babies are born on Mondays and Tuesdays than on weekends, which further complicates the problem. Depending on the group of people and how evenly distributed their birthdays are, the results can vary widely. But most of the time, you'll still have some very good odds.

But there is at least one highly practical application for this numerical phenomenon: computer hacking. There is a classic cryptographic computer attack known as the "birthday attack" which exploits the math of the birthday paradox. Using this method, a programmer can store the results of the birthday math in memory to decrease overall processing time when doing certain computationally useful things, such as attempting to crack a digital signature.Another thing that I discovered in my research is that a one followed by fifty-one zeros is called one *sexdecillion*. I knew those mathematician guys were hiding something in those big numbers.

As much as Math would like us to think that it is an advocate for structure and intuition, every once in a while it churns up something dastardly and unintuitive like the Birthday Paradox, the Monty Hall problem or Benford's Law. And we have no choice but to obey these fickle whims of the great control freak. But every once in a while, I like to divide by zero, just to show Math that I'm not powerless to retaliate.

To those who would claim that only a fool would fall prey to the Birthday Paradox, and that the true nature of the odds is perfectly intuitive, I ask this of your Rainman-like grasp of numbers... why is it that all of the totals in this article's first chart (aside from zero) end in the digit five? That outcome surprised me, but I currently lack the conviction to pursue the matter. I now see that numbers represent all that is soulless and wrong.

The numbers in the first column of your chart are powers of 365, that is 365, 365 squared, 365 cubed etc. 365 is an odd multiple of 5 (ie. 5 times some odd number), and since multiplying an odd multiple of five by another gives yet another odd multiple of five, so are all the other numbers here. An odd multiple of five is an even multiple of five plus another five, which is therefore a multiple of 10 plus another 5. Multiples of 10 end in a 0, and adding the extra 5 gives the last digit as 5!

I had heard in the past that the odds are more like 1 in 28, but it could depend on radical variables right down to the region one lives. Plus, you have people that try to give a child a specific birthday (specifically New Year's Day and Christmas) and that complicates the math even further. To add an interesting bit of reality... my birthday is Feb. 24th. Anyone else?

Birthday distribution isn't governed solely by logical and rational factors. There's also biology ("that time of month"), sociology ("not tonight, dear"), and heck, even climatology sometimes ("it's too hot to have sex tonight"). Those messy variables are enough to cock up (NPI) any distribution, it seems to me...

Climatology affects things on the other side too. Ask any ER physician or obstetrician - more women will go into labor during a thunderstorm. Also during full moons, but that's not climatology, strictly speaking.

I got a dose of this in a mathematics class once. Then the professor tried to prove to us that all horses are white.

A perfect indication that statistics is not maths :) I can see another way to approach this - Why can't you calculate by finding "The probablity of having two or more birthdays on this arbitary day is..."

I remember asking my math teacher back in the day to explain "i", the variable that equals the square root of negative one. It shouldn't exist, but it is used all the time. The teacher told me not to think about it and just use it. I could not not think about it! It held me up because I couldn't solve a logical equation and set aside contemplating some magically witchcrafted non-number. "What's in this soup?" "We don't know, nobody knows! Just shut up and eat it!"

karphi said: "I remember asking my math teacher back in the day to explain "i", the variable that equals the square root of negative one. It shouldn't exist, but it is used all the time."

Such numbers are fittingly named "imaginary" numbers.

RichUK said: "The numbers in the first column of your chart are powers of 365, that is 365, 365 squared, 365 cubed etc. 365 is an odd multiple of 5 (ie. 5 times some odd number), and since multiplying an odd multiple of five by another gives yet another odd multiple of five, so are all the other numbers here. An odd multiple of five is an even multiple of five plus another five, which is therefore a multiple of 10 plus another 5. Multiples of 10 end in a 0, and adding the extra 5 gives the last digit as 5!"

Yeah, definitely.

Furnace said: "To add an interesting bit of reality… my birthday is Feb. 24th. Anyone else?"

My birthday is the 26th...close!

There is an uncomplicated conclusion that can be drawn from this article: Math sucks.

Ps: Ten more days-- Feb. 20th-- until I am twenty-one. I rule.

Here is an intuitive way to think of it:

What are the odds that

no twopeople have the same birthday? Well, order the people. Then the first guy has his own birthday. For the second guy, there are only 364 choices out of 365, the second guy has only 363/365, and so forth. So the odds of forty people having unique birthdays is like364/365*363/365*...*326/365

We can approximate this by taking the middle factor and raising it to the fortieth power, so we get about (345/365)^40, which is very small because of the large power to which it is raised. So the chance of two people sharing is just 1-(345/365)^40, which is near to one, that is, there is a very good chance of having it happen. (Incidentally, that works out to about .895 or 89.5%, very close to the 90% quoted in the text.)

Actually, I think that any factor that skews the birthdays towards any uneven distribution will actually strengthen this law. For example, does more people being born in springtime increase the chance that two people in any room will have the same birthday?

Actually this is sort of interesting... perhaps I'll see if I can come up with a formal proof for it some time.

Well, yes, so long as the people are selected at random from a population, changing the probability of being born on different days from a 50/50 chance will increase the chance of coincidence. For example, say you have two days, and the probability of being born on the first of the two is p1 and the second p2. The chance of a coincidence on either day is p1*p1+p2*p2=p1^2+p2^2=2 p1^2 (Substitute p1 for p2 because the chance is equal). Now changing p1 and p2 so they still add up to the same value, you get the probability of coincidence (p1+x)*(p1+x)+(p2-x)*(p2-x)=p1^2+2p1x+x^2+p2^2-2xp2+x^2=2pi^2+x^2. Because x^2 is always a real number, the term x^2 is positive and thus a uniform distribution in fact causes the least coincidence.

Speaking of x^2 being always positive, the other topic of discussion that crept in here, complex numbers, is quite interesting. Recall grade school when you came across fractions. A simple fraction a/b is a solution to the problem a=bx, when finding the value of x (if b does not divide a evenly). It is not a "number" by any definition you had before that. And later on, when you learned of irrational numbers like sqrt(2), they did not fit the previous definition of numbers. So too is i a mathematical solution to a number question that was previously unsolvable. The really cool thing, though, is that if you assign a letter to i and do some operations on it, you get amazing consequences. For example the fascinating formula e^(pi i) + 1=0 and the beautiful Mandelbrot set. In fact, there are more complex systems of imaginary numbers than just i, such as the quaternions, which use i,j,and k. But to define them in unique ways, even more of your familiar mathematical laws have to go out the window.

Great article. My network security class was just talking about this the other day, and today(2-10) is my birth day....weird...

For the curious, why the number in the second column is always 5 (except for the first and last entries):

RichUK said that the number in the first column is a power of 365 and that it always ends in 5. If we subtract from that the number of ways for no two people to have the same birthday, we should get the number in the second column.

According to dmwit, the number of ways for no two people to have the same birthday equals 365*364*363*(lots more numbers). As long as we have at least two people, we'll have 365 and 364 in that product, which will make it a multiple of 10. (And if we have more than 365 people, then there are no ways for no two people to have the same birthday, and zero is still a multiple of 10.)

Since the number of ways for no two people to have the same birthday is a multiple of 10, then subtracting it from the number in the first column will leave the last digit (5) unchanged. So as long as we have at least two people, the last digit of the number in the second column is 5.

February 24th here!

What's UP with that?

Probability is about counting things. Counting is pretty easy, so long as you're careful to count the right thing, and those blasted formulas just simplify the counting a little. Say you want to know the odds that any 2 people in a group will have the same birthday (and let's assume they're evenly distributed even though they are clumped towards the spring). It's easier to count how likely they are to not have the same birthday: the first person in the room can have any birthday at all, the second has only 364 possible days without conflict, the third has only 363 days, and so forth. But consider that the people didn't choose their birthdays - they were all picked at random from the same set of 365, not from that constrained set. Coincidences are bound to arise.

What, then, does the table count? Certainly not the same thing. It's counting the ways in which birthdays of individual people can be arranged - person A could be born on any of 365 days, person B on any of 365 days, and person C on any of 365 days. True enough, but it accounts for people as individuals rather than considering them as a group. In fact, the birthday paradox formula accounts for the individuals, too, but the denominator cancels them out.

5318008

Nice. I like how your calculator thinks.

karphi said: "I remember asking my math teacher back in the day to explain "i", the variable that equals the square root of negative one.

When I was learning algebra I KNEW I was being hornswaggled somehow. When we got to the "i" thing I found the bridge they were selling

Feb. 22.... how weird. 20th, 22nd, 24th... what are those odds?! So should we look forward to an article about a math proof that shows our nonexistance? hmmm... maybe that was just a D&D thing.

and you're right, math does control us. that's why i pray in numbers.

i know this is extremely hard to believe, but i was born on 2-18-85. i turn 21 on saturday. friday night, 1 min after 1159pm.

so, we've got 18, 20, 22, 24th?

my guess: we were all more attracted to this article because our birthdays are coming up

"My birthday is the 26th…close!"

18, 20 ,22, 24, 26.

karphi said: "I remember asking my math teacher back in the day to explain "i", the variable that equals the square root of negative one. It shouldn't exist, but it is used all the time. The teacher told me not to think about it and just use it. I could not not think about it! It held me up because I couldn't solve a logical equation and set aside contemplating some magically witchcrafted non-number. "What's in this soup?" "We don't know, nobody knows! Just shut up and eat it!""

Electrical power uses a lot of complex numbers (which don't exist). They are very helpful CONCEPTUALLY.

Was everyone here born in February?!? Adding another real-world example that I forgot to mention: I work at a company with 34 employees and there are two separate dates where people share birthdays, which doesn't include the identical twins working there.

Cool website! To further fuel the February conspiracy theory fire, my birthday is on February 23rd.

Love the way you wrote this article, sure made me laugh a few times...

Anyway, in trying to make sense of it all, could someone please explain, IN ENGLISH ;o)

Why 1 person creates 365 Possible combinations of birthdays (which makes sense to me) but,

Why 2 people would create 133,225 Possible combinations of birthdays??? (I have no idea)

The way I'm seeing this at the moment (at the risk of sounding foolish) is that regardless of

wither there be 2, 4 or 8 people in a room, there are still only 365 days in a year...

What I need is a breakdown of how this is calculated.

MAKE THE HEADACHE GO AWAY!!! HELP!!!

PS: OK! so I'm no maths genius, but I am sure interested in how this works ;o)

"I now see that numbers represent all that is soulless and wrong."

Exactly...equations are the devil's sentences.

If I had to wager a guess, I'd say so many people responding have birthdays in February because it is February and when they saw a link to the Birthday Paradox they were more likely to be interested than people whose birthday is less imminent.

FWIW, my daughter's birthday is Feb 20th, but mine is in April. Hail to the Math nerds!

PS: OK! so I'm no maths genius, but I am sure interested in how this works ;o)"

basically you can except that when you are dealing with probabilities you just multiply a whole bunch of numbers. Or you can go around thinking about it and looking for examples until it gradually sneaks into your intuiton.

Ill give a couple of examples.

take a 6 digit license plate for simplicity sake we will say each character can only be a digit 0-9 no letters.

if we have just one digit then we have ten combinations 0-9. If we have have 2 characters we have one hundred combinations 0-99

3: 0-999 1000 combinations 10^3

4: 0-9999 10000 combinations 10^4

5: 0-99999 100000 combinations 10^5

6: 0-999999 1000000 combinations 10^6

you just multiply the number of combinations for any one character by the number of combinations for any other character. 10*10*10*10*10*10 or 10^6.

if you had letters too, then each character would have 36 possible combinations 0-9 +a-z =36 then you calculate 36^6 = 2176782336 possible combinations.

And Briefly there is the binary number system which i find to be a simple demonstration of this concept but you might not. As you may know the binary number system is base 2 meaning each digit can only be a 0 or a 1.

in binary you count like this

0,1,10,11,100,101,110,111,1000

with one digit there are two combinations 0 or one

0 1

with two digits there are four combintations 2^2

0 0

0 1

1 0

1 1

three digits make for eight combinations 2^3

0 0

0 1

1 0

1 1

1 0 0

1 0 1

1 1 0

1 1 1

and so on

with simple probablity it is always (possible combinations for any one variable)^(number of variables)

hope i helped... if i didnt i just wasted an aweful lot of time at work

joe:

the reason that there are 133,225 possible combinations for 2 peoples birthday is fairly simple.

person 1 can be born on 365 days, and person 2, likewise on 365 days.

so the number of combinations is number of ways to choose person 1's birthday times the number of ways to choose person 2's birthday, or 365 *365 = 133,225

may 31st here

I don't think the Birthday Paradox is that complicated to understand.

First, let's get away from the math for a bit. What if there are 367 people (curse you, Feb 29) in a room? What then is the probability that two of them have the same birthday? Why, it's guaranteed that two do! The "pigeonhole principle" says that if you keep picking unique birthdays for folks in the room, you will run out of unique birthdays before you run out of folks.

Now, let's do a tiny bit of math. Suppose there are 120 folks in the room and they all have unique birthdays. Then what are the odds that a 121st person with a randomly selected birthday has the same birthday as one of them? Well, about 1/3...1/3 of the unique birthdays are taken, and the other 2/3s aren't. That's pretty good odds, still.

But now think of it this way...imagine we start with 120 unique birthdays in the room, and add people #121..#130. Each of these folks has a roughly 1/3 chance of colliding with the original 120. So now you ask, "what are the odds that in 10 tries you don't roll 1 or 2 on a six-sided die?" In other words, what are the odds that you fail to achieve a 1/3 probability in 10 tries? The answer, it turns out, is 2/3 to the 10th power, which is a little less than 2%. If this is counterintuitive to you, get a die and try it. You'll likely get bored before you get through 10 rolls of just 3..6.

So it seems clear that if 130 people were in the room, two would almost certainly have the same birthday, because even if the first 120 didn't contain such a pair, one of the next 10 would. Now, it should all be coming clear. Imagine that instead we'd started with 110 people in the room and worked up to 120. By exactly the same logic, the odds would be almost identical. So in a room with 120 people, it's

stillalmost certain that two will have the same birthday.You see where this is going. Since with two people in the room, there's little chance they'll have the same birthday, the number of people for which the chance of a same birthday is 50% is going to be somewhere between 2 and 120 (already less than half), but by our math it looks way closer to 2 than 120. If you're mathematically sophisticated, you can now write down the right equation to get the number of people achieving a 50% chance. If not, you can look it up.

Hopefully, this is clear. I think I didn't make any giant leaps. Let me know if I am wrong.

I made a computer application that helps people understand the paradox. It's on my website at http://www.pushnshove.com/index.php?page=1

I hope this helps some people understand it.

The word sexdecillion has absolutely no more "dirty" meaning than the word "sextant".

"Sex-" is simply a Latin-derived prefix that means "six", and "deci-" means "ten", as we can probably guess from our regular interactions with it. (e.g., the deci-mal system)

Million means 1,000 times 1,000.

Billion means 1,000 times 1,000 twice---"bi-" meaning "two".

Sextillion, then, means 1000 times 1,000 ^ 6, which is 10^21 or 10 with 21 zeros after it.

Sexdecillion, therefore, is simply a mangled Latin-inspired way of saying 1,000 times 1,000 16 times.

If there's anything dirty about the name, it'd be its liberal "borrowing" of grammar and vocabulary from dead languages.

You all probably have heard of this one.

But if you look at the year you are born and then the year your father is born (or mother). When you reach their year in age (for example if they were born in 1933, then when you reach 33), they will be the year you were born.

I thought this was kind of cool. I think it breaks somewhere, but I think it holds true for all those up to 1999.

Try it.

Carlos

(Of course if you were adopted or otherwise not born of your parents then this wouldn't work).

"I ask this of your Rainman-like grasp of numbers… why is it that all of the totals in this article's first chart (aside from zero) end in the digit five?"

Anything ending with 5 multiplied by something else that ends in 5, produces an answer that too ends in 5

5x5=25

5x15=17

5x25=125

15x15=225

25x25=625

35x25=1225

45x45=2025

55x55=3025

and so on ....

If you look at 5^2, 25^2, 35^2 ...... n^2 where n ends in 5, you'll notice that the results are in a pattern

a = (n-5), b=(n+5), x = (a*b) + 5^2

OR

a=int(n/10), b=int(n/10) + 1, x=a*b + "25"

Suppose n= 65, then a=int(65/10) -> int(6.5) -> 6, b=int(65/10) +1 -> int(6.5)+1 -> 6+1 -> 7

x = (a*b) + "25"

x=(6*7) + "25"

x=42 + "25"

x=4225

18, 20 ,22, 24, 26.

WOW. My birthday is February 21st. What's the deal? One of your theories is that we're all interested in birthdays because ours is coming up...then why are there no February 12ths? And why has one SINGLE person not been may, or december, or august?

Maybe that whole Zodiac thing isn't a crock, after all. We all know Februarians are brilliant.

:)

My birthday is April 28th.

February is going DOWN!!

My birthday's on February/25. Cool huh?

So my hunch is that events we might consider 'synchronistic' are simply part of a larger mathmatical statistical model. So that old friend I bumped into the other day after thinking about her minutes before is not psychic, but just a piece of the pattern.

I was recently examining this problem, and was looking for information on the actual distribution of birthdays. All I could find was data from 1978 used by Geoffrey Berresford in the following article: "The uniformity assumption in the birthday problem, Math. Mag. 53 1980, no. 5, 286-288." A copy of the data set can be found here: http://www.dartmouth.edu/~chance/teaching_aids/data.html

Interestingly, this data directly contradicts Alan's contention that more babies are born in spring time. Many others here have made claims about birthday distributions, but I haven't seen anyone reference data to back them up. Where are you getting this information? I'd really like to look at the data.

For all the Feb birthdays.. since this link is on digg it's getting a fair amount of traffic. So even though a small percentage of all the Feb birthday visitors actually post a comment, it still looks like a lot of "coincidental" Feb birthdays. My birthday being in August, normally I'd be one of those people that didn't post. But I have nothing else better to do, so there ya go :)

FishSpeaker said: "Interestingly, this data directly contradicts Alan's contention that more babies are born in spring time. Many others here have made claims about birthday distributions, but I haven't seen anyone reference data to back them up. Where are you getting this information? I'd really like to look at the data."

Here's the citation of the more-babies-in-springtime reference, may be tricky to find:

Klamkin, M. S. and Newman, D. J. "Extensions of the Birthday Surprise." J. Combin. Th. 3, 279-282, 1967.My birthday is February 22nd as well, so that makes for at least one matched birthday, as long as I understand the comment history. So, how big is our "room", I have a feeling that several thousand have seen this page, so 1 match is not too surprising. Still, you gotta love it when an article comes along that can help enlighten the rhetorical question: "What are the odds of that?!?"

wingie said: "Sextillion, then, means 1000 times 1,000 ^ 6, which is 10^21 or 10 with 21 zeros after it.

Sexdecillion, therefore, is simply a mangled Latin-inspired way of saying 1,000 times 1,000 16 times."

Please see: Sense of Humor

FishSpeaker said: "I was recently examining this problem, and was looking for information on the actual distribution of birthdays. All I could find was data from 1978 used by Geoffrey Berresford in the following article: "The uniformity assumption in the birthday problem, Math. Mag. 53 1980, no. 5, 286-288." A copy of the data set can be found here: http://www.dartmouth.edu/~chance/teaching_aids/data.html

http://www.oregon.gov/DHS/ph/chs/data/finalabd/99/birthmo.shtml

was the best i could find.

goes by month, which..well..isnt all that helpful.

in better news, my 21st birthday is in a week. a friend at work turns 21 on the 17th.

Interestingly, this data directly contradicts Alan's contention that more babies are born in spring time. Many others here have made claims about birthday distributions, but I haven't seen anyone reference data to back them up. Where are you getting this information? I'd really like to look at the data."

I didn't even mention my own dramatic example of birthday intersections in the article: my wife shares a birthday wth her stepsister. After my wife and I got married, we learned that I share a birthday with that stepsister's husband. My birthday and his were not only the same day, but the same year. Crazy stuff.

I was wondering something. Me and my roommate met over the internet. the day we moved in we found out that not only do we have the same birthday, march 13th...but we were both born in the same year, 1979. so we both have the same EXACT birthday...what is the probability of that happening?

Just a quick thanks for your feedback guys ;o)

Hammy you're not from New Hampshire by any chance, are you?

My little bro turned 21 yesterday. Lots of people turning 21 this Feb!

To contribute to the maddness and further prove this insanity:

My birthday is Feb 18th.

Spock one said that a sane man in an insane society would seem insane. Since math seems to be the logical yardstick of, well, everything, I wonder which side of the fence we're all on.

ryan97ou said:

Ok... let's count this one up, and ignore the leap years to make things easier. What are the odds of you both being born on 3/13? Well, the odds of you being born on 3/13 are 1 in 365, but you walked into the room with that birthday, so it was a foregone conclusion and gave the date significance, so that much is a probability of 1. The odds of your room mate having the same birthday (nevermind year), are 1 in 365 (assuming evenly distributed birthdays).

To answer the question of the same year, again, it's a foregone conclusion that you were born in 1979, so those odds are 1. What are the odds of the mate having the same birth year? Probably pretty good. It depends on a zillion factors - how you met the person and life circumstances. Let's assume you put an ad in the paper and took the first caller. Let's assume the caller would be at least 18 years old, and not more than... (and here's the whopper of a guess) 40. The ages of people with room mates in the United States (where I am - but I don't know where you are), probably favors the 18-30 group. Others tend to get married or get their own places. An octagenarian might have called you up, too, so we shift the guess a little to the higher end. So... assuming (wild assumption) that the years are evenly distributed amongst the ages 18 and 40, birth years are from 1965 to 1987, inclusive, a 23 year span. We'll say, then, that there was a 1/23 chance of your having the same birth year as your roommate. A more complicated approach would probably show that it was more likely for you to have a similar birth year, with odds possibly as good as 1/6...

So 1/23 (odds of same year as you) * 1/365 (odds of same day as you) = 1/8395.

Bear in mind that my estimates of odds for your birthday collision are pessimistic. An optomistic answer would consider that birthdays are clustered towards the spring... oh, a 1/200 chance of March 13, we might say... And we could suppose that your roommate would be within 4 years of your age for compatibility's sake, reducing to 9 the number of possible years... 1/9 chance... and if those estimates are close, the 1/1800 odds aren't so long as you might have imagined. (Of course, for those born in August, the odds are not so good.)

To test it: survey at least 30,000 roommates in different dwellings and ask them if they have the same birthdate. Then you'll have an imperical estimate ;-) Good luck!

I remember not too many years ago a lottery number came up "0 0 0 0" in a game where you pick four digits from 0 to 9 in order (obviously, for the probability pedants reading, with replacement). When someone actually won with that number, a reporter asked of a lottery person, "what are the odds?" She didn't know! Well... there are 10,000 possibilities with each as likely as the others, so the odds were 1 in 10,000. You'd think the lottery folks would know that! Your odds of having met the roommate with the same birthday are undoubtedly better than your chance of winning at that particular lottery game.

Eric Leeson said: "There is an uncomplicated conclusion that can be drawn from this article: Math sucks."

There is an uncomplicated conclusion that can be drawn from this article: Math rocks.

The probability-specialized professor at my university explains this paradox to students each year in the introductory probability course. It's just one of those things that make math so cool.

After he shows the result, he goes on to talk about the hat-pile problem;

A bunch of college students are having a frat party, and they all throw their hats in the same pile at the door of the house when they enter the party. Now, if each and every one were to randomly pick one hat out of the pile on the way out, what are the odds that exactly one ends up with their own hat?

Incidentally, my brother's birthday is the 22nd of February. Mine is June 8th.

b-day: December 14

Just had to add my little grain of salt into this intellectual soup. Obviously I hardly understand anything that's going on within the article nor the explanations nor the little application that i downloaded to explain the b-day paradox. I never really understood my math teachers either. I see a pattern though. When one explains abstract mathematical concepts in the most simple terms, I only get more confused. There's a part of my brain that went out for lunch, won the lottery with the numbers 1,2,3,4,5,6,7,8 then fell in love with someone born on the same day, got struck by lightning while being attacked by a shark, and never came back. What are the odds of that one? I dare you to answer.

I also believe that mathematics is the purest language, probably that's why it is beyond my reach. Great article though, great responses, but how would i know, i hardly understood anything, it's just a hunch. I did understand the whole "sex" episode and then you lost me straight away when the numbers came in (just kidding, i'm not that dumb).

PS: I did really good in geometry though, but i suppose thats for kindergardner's. Well I'm still proud of it. Simple pleasures for simple people as they say. And where is JustAnotherName to tell us something relevant about the bible when you need him? A day is far from complete without a good dose of JV-ism.

Evil people! You've sent me on a quest to analyze all the birthday numbers I could find! And I don't have time! Argh.... Anyhow, I analyzed the aforementioned 1978 birthday numbers. There is a weekly pattern (presumably fewer births on the weekend, but I didn't look at that in detail). There is also a gradual trend towards more births as the year goes on (as one might expect with population increase). What surprised me was the lack of births in spring. January through March are slightly below average (about -0.6 sigma (sigma is 1 standard deviation - that is, 1 sigma is the amount of change we will ordinarily see when looking at this data set), but July, August, and September are gangbusters at about +1.2 sigma! December had a short spike, too. Apparently it was too hot in the summer of 1977, and when the weather cooled off, people decided they needed to get together. The hearsay wisdom that birthdays are clustered towards the spring (which might hold true in other years but certainly didn't hold true in USA 1978) is based on the notion that mothers prefer to carry their babies to term in the less-than-sweltering spring months, but the 1977-8 family planning doesn't seem to have been so well coordinated. I made my spreadsheet in OpenOffice.

PDLagasse said: "Birthday distribution isn't governed solely by logical and rational factors. There's also biology ("that time of month"), sociology ("not tonight, dear"), and heck, even climatology sometimes ("it's too hot to have sex tonight"). Those messy variables are enough to cock up (NPI) any distribution, it seems to me…"

Add in the rating of the movie you just watched and "bam"...you get an imaginary number.

Great article..I learned something and now I'm scared of math....again after years of therapy (college).

PDLagasse says:

Ah! That reminds me... we can assume (quite reasonably) with a population of several million people that "that time of month" will be uniformly distributed throughout the women. Sociology likewise has a near-uniform distribution, but climatology does not, to be sure. Snowstorms often precede a burst of births - but since snowstorms are local events, the local hospital sees it, but the nation of millions doesn't get much of a blip out of the event. I hypothesized this morning that the births in 1978 summer-autumn were encouraged by the hot summer of 1977 (no, I didn't look up the temperatures) and the ensuing passion when autumn 1977 came along. Of course, other factors will cause dips and surges, but few are on a national scale. I'm curious to know what happened to births in June 2002, for example. I bet it dips because the events of September 11, 2001 gave people pause and knocked romance from the air for awhile.

Statistics is math's ugly little brother, Staticians are not Mathematicians (wich is why some people consider the poisson equation to be rather fishy).

Now if you dont mind i need to solve Tanh(e(2pi) + 3pi/4i), my complex number class midterm is friday

Ralph and Carolyn Cummins had 5 children between 1952 and 1966, ALL were born on the 20 February.

CRAZY!

It's super simple. If you have a deck of cards with 365 unique cards in it and you start to flip over cards one at a time, you aren't looking for a match to one specific card, so each card turned over adds a whole new possibility to match to. With each card turned over you have more and more possible targets. Remember, you're looking for any two random matches, you're not looking to match one single card out of the 365. It's simple.

Another way to put it is this; What you are thinking about in the situation you are presenting really is asking what are the chances that a group of people will match YOUR birthday. Those chances are 1 in 365. By adding more people you are adding more possible birthdays that any 2 people can match. See what I'm saying?

Nth Degree you have once again proven my theory :

[math --> simplicity = indra c.--> utter confusion]

Or in lay terms: The more math is simplified, the more i get confused.

And the "you see? It's really simple" by god that just puts me out of my sane self. I will now go bite a wall face-on and curse you mathematicians! Have you ever tried biting a wall face-on? Well there you have it, that's just about the same relationship i have with the simplicity of mathematics. When I think back to my math teacher who pulled off the "you see? it's really simple" countless times, i feel really really sorry for him to have had me as a student. No Harvard degree will ever prepare you for the complexities (or not) of a brain that's on strike.

On the flip side i do understand the theories and paradoxes of quantum physics as long as none of it is "simplified" by numbers (of course). That would really be evil. If you really want to insult me, put candles on my birthday cake and ask me to count them.

So what are the damn odds that the person who commented second has the same birthday as me?

Feels like this happens all the time.

Feb 24, by the way.

Heheh...

# of people, 9, 41969002243198805166015625

In my school of 3500 students, I am the only one with the birthday of March 14. What is the probability of that occuring?

It's probably a good idea to think about everyone whose birthday you know (family, friends etc, you can include celebrities if you wish). There are probably around 300 people I know whose birthdays I have heard at some point (not including celebrities in this case), yet at least 5 of these were born on October 2. I also know 2 people from those 300 or so who share my birthday (October 24), although this is obviously influenced by the fact that I am more likely to hear and remember people's birthdays when they are the same as mine.

lahuard said: "In my school of 3500 students, I am the only one with the birthday of March 14. What is the probability of that occuring?"

Probably higher than you think (don't know how to work it out exactly though). In a school of that size, it is very likely that no one has their birthday on a particular day, let alone one person. Admittedly the probability is much lower when considering only one person.

Just so I feel included I'm Jan. 9 ....I know no one cares. But my sister is feb. 27

My birthday is Feb 22 also. So cool.

I have read all the posting (well the ones i could understand some just went way over my head). I would like to know the likely hood of 2 ppl sharing the exact same birthday 11-03-81, but before finding that out had known each other for most of their lives. We never saw each other outside of events at a certain Uncles place, so we never knew the fact about our birthdays until the year we turned 19.

I was siiting in the waiting room at the Dr.'s office waiting for an appointment for my son. There were a total of nine people in the waiting room. There were two women with infants, so my wife being curious, asked them how old their children were. The children were both six months old and one of the mothers said that their child's birthday was May 24. The other said that it was amazing, her child was born on the same day. My son who was five years old was also born on the same date, May 24! It gets even stranger... The father of the fourth child says that he can't believe it but his birthday was also May 24!! So we all got out our ID's and proved the coincidence. It was very amazing in the fact that four of the nine people had exactly the same birthdate.

Joe:

With two people, the odds are that high because of all the different combination. If you made a table of them, it would go like this:

Person 1 Person 2

Jan 1 Jan 1

Jan 1 Jan 2

.... (362 more lines)

Jan 1 Dec 31

Jan 2 Jan 1

.... (363 more lines)

Jan 2 Dec 31

You would end up having 365 * 365 lines in your table.

As for the late February "coincidence," how many people do you think have read this page? Probably thousands. Most people saw a birthday that wasn't near their own, and didn't bother to respond. A few people saw a birthday that was near their own and thought "Wow! That's neat! I'll type a response about this amazing coincidence!"

Finally, my own birthday coincidence story: My name is Dan, and my birthday is August 30th. My first wife's birthday was August 31st. My boss at the time was also named Dan. His birthday was August 31st. His wife's birthday was August 30th. When this came out, he didn't believe me until he'd seen both our IDs.

Daniel Lew said: "I got a dose of this in a mathematics class once. Then the professor tried to prove to us that all horses are white."

How exactly did he try to prove that....? I've seen brown horses.... Was that supposed to be sick humor?

c_s_1987 said: "... my birthday (October 24)"

Oh... so close... October 25 for me.

And Jim Roepcke is probably right about why so many of the original posters had birthdays around the time this article was posted: their birthdays were coming up and they were paying more attention to events/articles/etc that had birthday references (like this article).

ke4roh said: "PDLagasse says: I'm curious to know what happened to births in June 2002, for example. I bet it dips because the events of September 11, 2001 gave people pause and knocked romance from the air for awhile."

I would think it would be the opposite. I know a lot of people that increased their mating patterns shortly after that day - myself and wife included. In times of severe tragedy, humans have a tendency to grope for any form of solice they can find. For those couples of viable breeding capability, with feelings of "I'm at a loss for what to do... just hold me", the situation was ripe for lovin'. I'd suspect June 2002 had a slightly higher than typical birth rate (in the USA at least).

PDLagasse said: "Birthday distribution isn't governed solely by logical and rational factors. There's also biology ("that time of month"), sociology ("not tonight, dear"), and heck, even climatology sometimes ("it's too hot to have sex tonight"). Those messy variables are enough to cock up (NPI) any distribution, it seems to me…"

Although you're correct, it actually makes it

morelikely that you will get a match.Everyone knows that a coin flip is 50/50, and so the chance that you will have 2 flips the same is 50% (HH,HT,TH,TT - 2 times out of 4). Now consider a biased "coin flip". Roll 1 dice, and look at the result as either "6" or "not 6". If you roll 2 dice this way, the chance that they will both give you the same result is around 72% (26/36 if you want to be exact). You can extend this by doing "1,2" "3,4" "5,6" (12/36) then "1,2,3" "4,5" "6" (14/36) and "1,2,3,4" "5" "6" (18/36). You could go even further if you have bigger dice.. but I think that proves my point.

If you want to do an experiment based on this paradox, tell people abbout this paradox, and then ask them how many 20-sided dice they would need to roll so that a pair comes up at least half the time, and then count the number of people that say less than 6... (And you can include me in that, because I guessed 5)

Oh, and for those that care, August 27th

Emmy says:

Daniel Lew said: "I got a dose of this in a mathematics class once. Then the professor tried to prove to us that all horses are white."

How exactly did he try to prove that….? I've seen brown horses…. Was that supposed to be sick humor?

Look up Horse Paradox in Wikipedia.

Wow. How come whenever im done reading these comments thats always what I have to say? You people are idiots. Now math tackles all non-living matter. Einsteins theory of relativity makes much sense. You can predict anything that will happen by knowing everything happening now because you use logic. But when it comes to people, logic entirely fails. Math is entirely based on logic. So math fails too. I don't know what retard scientists (if any) try to prove this probability thing with math, but they must be out of their minds. There are no set patterns in people. Patterns will never be constant as long as living things alter them. I don't understand why everybody is getting so hooked on this. Probability DOES NOT APPLY TO PEOPLE. It is like asking "What is the probability that someone will pick tihs over this?" What is the probability a couple will pick this night to have sex compared to this? You don't know the likes dislikes, if they like warm or cold weather, how their day went, so on and so forth. If you did you could predict. But this is a GLOBAL scale. No computer could handle this.

By the way. Those February birthdays are all together because people in February saw that someone else was born then adn wanted to put that because it was close. And consider the fact that this was posted in February. If it was posted in October, someone will have said that their birthday is coming up, and someone else will say "Mine too!" and that will snowball until everyone starts posting their birthdays and it will be shown that there is a wide diversity of birthdays.

ChickenHead said: "Oh… so close… October 25 for me.

"

c_s_1987 said: my birthday (October 24), "

This is proof right here. After c_s_1987 tells his/her birthday, Chickenhead tells his/hers because its close. The February coincidence was caused the same way.

MrDlCastle said: "You all probably have heard of this one.

But if you look at the year you are born and then the year your father is born (or mother). When you reach their year in age (for example if they were born in 1933, then when you reach 33), they will be the year you were born.

I thought this was kind of cool. I think it breaks somewhere, but I think it holds true for all those up to 1999.

Try it.

Carlos

(Of course if you were adopted or otherwise not born of your parents then this wouldn't work)."

Carlos you are an idiot. There is nothing special about that.

E.G. : I'm born 1991. My parents are born in 1958. When I was born, they were 33. That means that 33 years passed from 1958 to 1991. So 58 + 33=91. Now when you already have 33 (parents age when you are born) and you just add 58 (the years it takes for you to reach their number) you get 33+58 which is also 91.

Nothing special. Nothing complicated. Simple math. Think.

Dear Didoka,

If you are as smart as your snide comments seem to suggest, you would obviously know that when math is applied to people we call it something different. We call it statistics.

You are right, math does not really apply to people. There are exceptions, such as the "Magic Ratio" that stems from the Fibonacci Sequence, but we're talking about numbers, not geometry, so I'll leave that one alone.

But you are completely wrong if you think probability does not apply to people. Every company in the world that has ever advertised has used statistics and probability to solve the question of "what is the probability that someone will pick this over this" in order to make their ad campaigns as effective as possible. And these companies are able to predict which methods of advertising will be most effective for any given demographic, despite only knowing a small portion of data (called a sample) from the population of interest.

Statistics was created specifically to simplify the analysis of data, because it just isn't worth trying to make predictions by looking specifically at every piece of data. It's already been mentioned that births will be higher on Monday and Tuesday because hospitals don't want to deliver babies on the weekend (doc's gotta get his golf in). For you to say we still can't predict when most births will occur makes you look like the idiot, not anyone else on this page. And this is only one of an almost infinite number of situations that can be examined for patterns. I'm guessing you PROBABLY don't want to risk making yourself look like an idiot again by questioning every one of those situations.

I am sorry this thread was so far below your level of intellect that you had to force your negative attitude upon the rest of us, but in reality your failure to recognize that people are indeed interested in probability and that it indeed is a very useful thing to know about if you want to really be successful in this world shows everyone reading this that you really don't know what the hell you are talking about. Rather than embracing this thread as an interesting sociological experiment, if nothing else, and learning how to effectively interact with people whose minds operate in a different way than your own, you feel the need to project your insecurities on the people here who are simply enjoying learning about an unintuitive phenomenon.

Basically, what I'm trying to say is that you are clearly a loser, with a stupid user name and a bad attitude, and you probably haven't gotten laid in a long time. Which is ironic both because of the discussion of sex that has already come up, and also because the stupidest person posting on here has most likely gotten more ass to date than you will in your entire life. Sucks to be you.

Sincerely,

Dave

lahuard said: "In my school of 3500 students, I am the only one with the birthday of March 14. What is the probability of that occuring?"

Well, that depends on your exact question. What are the odds that in a school of 3500 people, someone has their own birthday by themselves (ie, no others share the birthday? For this, the probability is extremely high (I have no exact numbers, but another classical math. fact, the "coupon collector [problem]", says that, with only 3500 people, it's actually very likely that there are still some days where nobody was born).

If the question is more like, "what are the odds that among 3499 other people, none were born the same day [of the year] as I was?", then, assuming equal distribution of birthdays, the probability can be expressed as (364/365)^3499, which is pretty low (0.0067%), roughly one in 14000.

Now, many of these apparent coincidences ("hey, we're talking about the odds of something unlikely happening, and look, here's a very unlikely coincidence!") can be "explained" in two ways at least: first, if you look hard enough for coincidences, you will find some ("Hey, my name is Philippe, and my girlfriend's name is Marie-Line - out of all the possible names, what were the odds?"); and second, people will tend to come forward with "weird" stories much more easily than those with nothing "weird" to tell (which can also explain the sequence of February birthdays: once someone came forward with their own February birthday, anyone with a close birthday also came forward, while those with June or January just didn't intervene).

I've noticed the skew towards fall birthdays within my family. There are seven of us. I - second oldest - was born March 16th. My older sister was April 25th. Everyone else falls between September 5th and November 6th. (9/5, 10/3, 10/23, 11/4, 11/6).

Not a summer or winter birthday among us. That's what I call skewed.

Honestly, the math started to leave me behind. I don't see the whole thing as so unintuitive once you think about it. Let's say you go to a party and there are a bunch of people there. You ask the first person what their birthday is. Its not yours. Not strange. You go on to the second person and so forth. it wouldn't freak me out if the first twenty people were duds. If I hadn't found a match by person twenty or so, it would start getting kind of weird because life isn't that pefectly random. It would seem odd to me that after asking forty people I should have no matches. Not impossible but odd. Its like lottery numbers. Intuitively you'd think that since the numbers are chosen at random you should spread out your number choices(2,14,27,34,48,59) but in real life the numbers aren't that "random". You get stuff more like 2,4,7,15,18,42,49. For fun take a lottery ticket and fill out the winning numbers and look at the little visual patterns.

minor correction

"If i hadn't found a match by person forty or so it would start getting weird. "

The thing is I run into weird coincidences all the time, as i'm sure most do, but like I said, its because life isn't perfectly random.

Can this same theory be applied to something a bit more useful, like say, the lottery. If chances of winning are 1 in 55 million, do I need to buy 55 million tickets to be 100% sure of winning? Or do I have a 99.7% chance of winning with something significantly less, like 8 million tickets? Anyone?

Hello.

How are the numbers in the second colum (# of these combinations where at least two people have the same birthday) computed ?

*** It would seem odd to me that after asking forty people I should have no matches. Not impossible but odd. ***

Wow, you have an amazingly poor grasp of the problem, and probability generally. If you ask 40 people you will still almost certainly not have a match (odds just over 10%). You would in fact have to ask 253 people to have a 50% chance at even one match of your birthday. The (to some people) surprisingly high odds of a match with 23 people come not because any one person has much of a chance of a match (they don't) but because there are 23 people who could have a match.

If I were you I'd stay away from Vegas.

*** Intuitively you'd think that since the numbers are chosen at random you should spread out your number choices(2,14,27,34,48,59) ***

No, intuitively YOU'D think that, but that's because your intuition is for shit. You're somehow confusing RANDOM with EVENLY SPREAD OUT. Nothing could be further from the truth. Odds are some clustering will occur, because there are more combinations with some numbers clustered than combinations without.

It has nothing to do with "real life" not being "that random." It has to do with you having no idea what randomness really means.

*** Probability DOES NOT APPLY TO PEOPLE. ****

Wow, that's amazingly retarded. Somebody should inform the actuaries of the world that they're out of business.

*** Electrical power uses a lot of complex numbers (which don't exist). They are very helpful CONCEPTUALLY. ***

NEWS FLASH: No numbers "exist." They are all helpful CONCEPTUALLY. Complex numbers are no different than any others in either of these respects.

CONCEPTUALLY, Mark C sounds like a very dis-likable person.

I too share a hate of numbers............. if course my hate did just increase a sexdecillion times as a result of reading this. (argh so painful my heads gonna explode)

Conan Y Crom.

In J.R.R. Tolkiens' "The Hobbit" he mentions that when a hobbit has a birthday they give rather than get presents. Not a bad idea, you never know when you'll get a present. How many friends must you have in order to have a 99% chance of at least one present every day ? I don't know and wouldn't even attempt the math but I'd guess you need a very large phone book (NY City size at least) to keep track of all those friends. Numbers anyone ?

"I ask this of your Rainman-like grasp of numbers… why is it that all of the totals in this article's first chart (aside from zero) end in the digit five"

Because, all those numbers are multiple of 365.

Doesn't anyone here have at least a basic understanding of math?

Odd coincidence:

While in high school in Nebaraska (important later) one of my best friends had the same birthday as me, but two years difference. Fun coincidence. Fast forward to after graduating and getting a job. One of my coworkers had the same birthday. Funny coincidence. Then it turns out that we were not only born on the very same day (same age), but at the very same hospital. The odds of that aren't necessarily high. But this is where it being Nebraska comes in, since I was born in San Antonio.

BOOBIES!!!

My birthday is August 5th, and though I have met many people with birthdays on the 4th and 6th, I have NEVER met someone with that same birthday (aside from from famous people and people I never asked).

November 14th birthday

May 14th is my B'day,

Going a bit off the topic,

I see a lot more men on Damninteresting than there are female (going by Screen names atleast) , in a bit of sexist comment :- Does this mean that men have more Graymatter than females , or perhaps they are the ones with more internet access and a tendency to laze time away on the internet.

Allan, can you confirm from the registrations, if it is not too private to share, do we have more men here than women ?

The Birthday Paradox is but a tiny element of the probability of duplication (or collisions). The probability of duplication is a whole lot more important when applied to real-life situations. For example, what is the probability of two DNA sequences to be identical, when N persons are considered?

There is now software to calculate a wide variety of duplication cases or collisions, including the Birthday Paradox. This is the best material on duplication, birthday paradox – software is also available:

http://saliu.com/birthday.html

(The Birthday Paradox: Combinatorics, Probability of Duplication, Coincidences, Collisions, Lottery, Roulette, Social Security Number (SSN), Genetic Code, DNA).

Happy birthday, to Whom it may concern!

wow there was absolutely no point to the 2nd half of that, it's like u decided to just keep saying stuff to sound intelligent instead of just saying 5x another odd number ends in 5 and leaving it at that ahahahaha...and as for Brett, b4 ur post i found at least 3 ppl in may and u said not a single person was in may. burn. aaand the reason i found this was bc someone had a stupid facebook app that said 6% of the world had her birthday and i decided it was simply not true and I have been looking into it since.

and btw, this site doesn't work with google chrome, that needs to be assessed.

sradz113, I hate to point it out to you but there is absolutely no point to both of your comments. No one cares. (my comment is purely caring about you, not your comments. I think the appropriate way to finish this up in a way that you understand is "ahahahahaha" and "Burn, Pregnant Dog").

My birthday falls on the 19th of September and I actually know someone in my own town that has the same birthday. Guess we beat the odds...

I share my Birthday with my Grandmother. How's that for coincidence? To one up myself, I know two people with the April First birthday, and they're the same age and opposite gender.

Math is not convoluted, twisted, or other wise trying to control us. These seemingly counter-intuitive properties found in math seem counter to intuition only because common intuition is flawed.

don't do this very much, but i actually had to chime in, my father was born 10-29-1949 and i was born 10-29-1972, how odd is that? what is the probability of this? i,ve never met anyone else with father/son bday thing, so any advice on figuring this out or website etc etc would be appreciated

i was reading saliu.com but i thought my brain was gonna crash

btw mom was born july 19 and sis was april 11ish(i know i'm a horrrrrrrible bro.)

You also have to decipher the difference in percentage of men that have babies a year compared to the women that have babies.... quite complex math.

I came across this site today on 21st Feb. It seems that (sourced soley from this article) a lot of people with birthdays in February are into maths - did you know that Einstein was born on 22nd Feb (so was I incidentally) - so what about the probability of a link between maths geniuses and people born in February? ;)

My birthdays 24th Feb too ... rubbish with maths but there seems to be alot of Feb birthdays in here. Which begs the question .. what are the percentages of a person being a Pisces more likely to look into something like this?

If n people are present in a room, what is the probability that no two of them celebrate

their birthday on the same day of the year? How large need n be so that this

probability is less than ½?

An infinite sequence of independent trials is to be performed. Each trial results in a

success with probability p and a failure with probability 1 – p. What is the probability

that

i) At least 1 success occurs in the first n trials;

ii) Exactly k successes occur in the first n trials;

iii) All trials result in successes? (5)

Q4. (a) The probability mass function of a random variable X is given by

p(X = i) = cli i!, i = 0,1, 2,…, where l is some positive value. Find

i) P (X = 0);

ii) P (X > 2).

CO absorbs at 2.143 × 105 m-1 which is similar to the absorption of NO+ ion. Can yougive any reason for this similarity?

How do u get 290,299,465????

Calculating the probability that everyone's birthdays are different is much easier: product( (365-n)/365 ) for n = 0 to group size -1.

eg, for 23 you have:

(364/365).(363/365).(362/365). [...] .(344/365).(343/365) = 0.4927.

1 minus this value is the probability that at least 2 are shared = 0.5073

ie, in a group of 23 people, there is greater than 50% chance of 2 people sharing a birthday. Simple huh?