00:01
So we are told that there are 12 people at a party, and we are interested in the probability that exactly two have the same birthday.
00:23
So the way that we can answer this is we can say that the probability that an individual, the probability that an individual is born on day x is equal to one out of 365.
00:48
There are 365 days in the year, all equally likely, so a probability that an individual is born.
00:53
Born on a given day x is one out of 365.
00:57
And the probability that an individual, i'm abbreviating i and d, is not born on day x, is equal to 364 out of 365.
01:17
And as a result, this becomes a binomial distribution where the probability of getting, i'm going to say y is equal to the number of people born on day x, and the probability of getting y follows a binomial distribution where we take the number of trials, which is 12, and the number of people you can get out of 12, or sorry, the number of times they can get y people out of 12 times the probability that the person is born on day which is 1 over 365 to the yth power times 364 over 365 the probability that a person is not born on day x to the 12 minus y so we are interested in the probability that y equals two that exactly two people are born on day x and the other 10 people are not and so this is going to give us 12 factorial divided by 10 factorial times 2 factorial times 1 over 365 squared times 364 divided by 365 to the 10th, which is going to equal 12 factorial divided by 10 factorial times 2 factorial just gives us 12 times 11 times 10 factorial divided by 2 times 10 factorial, which just gives us 12 times 11 divided by 2 times 10 factorial, which just gives us 12 times 11 divided by 2 or 66.
03:03
So we get 66 times 1 divided by 365 squared times 364 divided by 365 to the power of 10, which is equal to 0 .000481996...