Here is a classic problem we must discuss in a probability class. Suppose that I

ID: 3027970 • Letter: H

Question

Here is a classic problem we must discuss in a probability class. Suppose that I select n people at random to attend a surprise party. Assume that every person I select has an equal probability of being born on any day during the year (365 days ignore leap year), with each day being equally likely. Find an expression for the probability that each person at the party has a different birthday (as a function of n). Using this expression, tell me how many people I need to select to ensure that the probability that two people share a birthday is greater than 50 percentage? This problem is considered a "paradox" because the answer is far lower than what most people would guess. Since the answer to this problem is widely known, be sure to take the time to really understand it and show your work. You may also be interested to know that a type of cryptographic attack is based on this problem: http://en.wikipedia.org/wiki/Birthday-attack

Explanation / Answer

here suppose first person has birthday on one date

so that second person does not share birthday with him has probabilty =364/365 as he can have birthday on remaing 364 days

similarly third person does not share birthday with them has probabilty =363/365 as he can have birthday on remaing 363 days

hence for n people not sharing birthday has probability =365.365 *364/365*363/365*....*(365-(n-1)/365

=365!/((365)n*(365-n)!)

therefore probability that at least 2 will share birthday =1-P(no one share birthday)

=1- 365!/((365)n*(365-n)!)

as we want probability that at least 2 will share birthday to be greater then 50%. hence

1- 365!/((365)n*(365-n)!)>0.5

365!/((365)n*(365-n)!)<0.5

by hit and trail we get n>22

that means we require minimum 23 people for having a probabilty more then 50% so that 2 people can share birthdays.

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Here is a classic problem we must discuss in a probability class. Suppose that I

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