Question 989837: There are 65 people in a room. What is the probability that at least two have the same birthday? Round your answer to four decimal places.
Found 2 solutions by Boreal, solver91311:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! It is 1-exp - {n*(n-1)/2)/365}=1-exp -5.6986=1-0.0034=0.9966.
(1-p^n)^-n becomes lambda as n becomes large (and np is constant)
e^-lambda is probability of not being possible. This is an approximation, but allows calculation without large factorial numbers.
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
If there are 2 people in the room, the probability that they DO NOT share the same birthday is  ,
For 3 people, the probability of not equal birthdays is 
For 4, 
And so on.
So for 65 people, the probability of NOT sharing a birthday is
The probability that at at least two DO have the same birthday is 1 minus the probability that none do.
The problem you are going to have is that you are going to have to calculate this by multiplying these 65 fractions rather than by the formula. Even Excel won't do above 170!.
If you have Excel, you can do it this way. In one column of a spreadsheet, enter the numbers 365 down to 301. In the next column, enter 365 sixty-five times. In the next column, enter a formula of the first column cell divided by the second column cell. Then use the PRODUCT function on all of the numbers in the third column. Then subtract the result from 1.
John
My calculator said it, I believe it, that settles it
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