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THE BIRTHDAY PARADOX

    If there are 60  people in a gathering, will any two persons have the same birth day?  We are doubtful. we think if there are more than 365 persons, there may be a possibility of two persons sharing same birthday.  But the statistics' answer is stunning.  Here we go.

      In a coin toss, chance of getting head or tail is 50 % or 1/2.  Chance of head is 1/2 and the tail is 1/2.  Hence the total probability is 1.  

      In a dice throw, chance of getting '1' is 1/6 and the chance of 'not' getting '1' is 5/6 = 1-1/6. 

      In a gathering, the chance for a person to have any date in a year as a birthday is 365/365=1=100%.  For another person, the chance of not having a birthday same as first person is 364/365.  For a third person, the chance of not having a birthday same as previous two persons is 363/365.  Hence the probability for not having a match (two persons having same birth day ) among three persons is (365/365) * (364/365)*(363/365)  (based on property of probability).

      In another words,  the probability of having a match among three persons is

                1 -  (365/365) * (364/365)*(363/365)    

 So the chance of finding a match of birthday among N persons is 

                       = 1 -  (365/365) * (364/365)*(363/365)* ........*(365-N+1/365)

                       = 1-[365*364*363*...........*(365-N+1)/365^N]  
  
Using the above formula, let us calculate the probability of a match for different number of persons N = 1,2,3... and tabulate the results.

 N                                P
no. of persons             probability of a match

 1                                 0
9                                 0.1 or 10%
23                               0.5 or 50%
30                               0.7 or 70%
58                               1    or 100%


     So the chance for two persons having same birthday (a match)  is 50% even for 23 persons.  A shocking result.  If there are 58 persons in a  room, there is a 100% chance for a match.Try to verify yourself.

      This birthday problem(clash of dates, clash of data, clash of occupancy).  Finds applications in spread of epidemics, photons occupancy and cryptography.    
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