THE BIRTHDAY PROBLEM

After a long time i decided to write on one of the famous problem on probablity

                                                The Birthday Problem
In the following problem,which is often called the birthday problem,it is required to determine the probability p that at least two people in a group of k people will have the same birthday,that is,will have been born on the same day of the same month but not necessarily in the same year.For the solution presented here,we assume that the birthdays of the k people are unrelated (in particular, we assume that twins are not present)and that each of the 365days of the year is equally likely to be the birthday of any person in the group.In particular,we ignore the fact that the birth rate actually varies during the year and we assume that anyone actually born on February 29 will consider his birthday to be another day, such as March 1. When these assumptions are made, this problem becomes very simple. Since there are 365 possible birthdays for each of k people, the sample space S will contain 365k outcomes, all of which will be equally probable. If k>365, there are not enough birthdays for every one to be different, and hence at least two people must have the same birthday.So,we assume that k ≤ 365.Counting the number of outcomes in which at least two birthdays are the same is tedious. However, the number of outcomes in S for which all k birthdays will be different is 365 P k  since the first person’s birthday could be any one of the 365 days, the second person’s birthday could then be any of the other 364 days, and so on. Hence, the probability that all k persons will have different birthdays is
P365,k /365^k. The probability p that at least two of the people will have the same birthday is therefore p=1− P365,k /365^k =1− (365)! /(365−k)!/365^k.


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