Odds of two people having same birthday

[DVO]

This doesn't sound right. Read it in an excellent book on Warren Buffett, who of course is very proficient in the area of probability.

In the book, Buffett is in a room with 25 other people, and bets a friend that among this group will be at least one shared birthday. Buffett wins the bet and informs his friend the odds were in favor of it.

Doesn't sound right to me at all. Any thoughts? calculations?

[Navers]

the first person's birthday is, let's say, Jan. 1st. The probability of the next person having the same bday is 1-(364/365), which is 1 minus the probability of them not having the same birthday. The probability of the person after that (3rd person) sharing a birthday with either of the first two is 1-(364/365)*(363/365). Basically the formula is 1- (364*363*362...340)/(365)^25. That comes up to .59824082. There's a 60% chance that if you're with 24 other people, someone has the same birthday as someone else.

[RocketManJames]

Do a search on "Birthday Paradox." It is a well-known problem, and I'm sure you'll find plenty of information on this. Not only does he have an edge with 25 people, but his edge is pretty large. If I recall, the cut-off is 23 people to make it +EV to bet that there is a birthday. Of course, this assumes equal chances of having any birthday. I have read that this is simply not true due to increased sperm counts in mammals during certain seasons. I suppose with non-uniform distribution of birthdays, the likelihood of two having the same birthday would increase(?) (not sure about that).

-RMJ

[tubbyspencer]

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Do a search on "Birthday Paradox." It is a well-known problem, and I'm sure you'll find plenty of information on this. Not only does he have an edge with 25 people, but his edge is pretty large. If I recall, the cut-off is 23 people to make it +EV to bet that there is a birthday. Of course, this assumes equal chances of having any birthday. I have read that this is simply not true due to increased sperm counts in mammals during certain seasons. I suppose with non-uniform distribution of birthdays, the likelihood of two having the same birthday would increase(?) (not sure about that).

-RMJ

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Yes, if humans are more likely to be born on some days than others, that increases the chances that any random 2 will have the same birthday.

[boogerboy]

Amarillo Slim writes about it in his book too. He took a proposition bet with I think 30 in the room.

[pzhon]

[ QUOTE ]
In the book, Buffett is in a room with 25 other people, and bets a friend that among this group will be at least one shared birthday. Buffett wins the bet and informs his friend the odds were in favor of it.

[/ QUOTE ]
The birthday problem is well-known. Here is a way to see the right order of magnitude for when birthdays typically start to coincide.

What is the average number of pairs of people with the same birthday? It is the sum over all pairs of people of either 0 or 1, with a 1 if the two have the same birthday. On average, each pair of people contributes 1/365 to the average number of pairs of people with the same birthday.

When there are at least 28 people, the number of pairs of people is over 365, so the average number of pairs of people with the same birthday is at least 1. In some groups of 28 people, there would be many bithday pairs, and in some there would be none, but if you expect there to be 1 then the probability of distinct birthdays is about 1/e = 36.8%. If you expect there to be n birthdays, the probability that all are distinct is roughly 1/e^n (by the Poisson distribution).

At 26 people, the expected number of birthdays is 325/365, and 1/e^(325/365) = 41.0%, so the probability two people have the same birthday is roughly 59.0%. The real probability in the uniform model is 1-(365/365)*(364/365) *...((365-25)/365) = 59.8%, not far off.