Suppose there are C people, each of whose birthday(month and day only) are equal

Question

Suppose there are C people, each of whose birthday(month and day only) are equally likely to fall on any of the 365 days of a normal (i.e.,non-leap) year.

(a) Suppose C = 2. What is the probability that the two people have the same exact birthday?

(b) Suppose C >= 2. What is the probability that all C people have the same exact birthday?

(c) Suppose C >= 2. What is the probability that some pair of the C people have the same exact birthday? (Hint: You may wish to use (1.3.1).)

(d) What is the smallest value of C such that the probability in part (c) is more than 0.5? Do you find this result surprising?

Explanation / Answer

Ans:

a)Probability that the two two people have the same exact birthday=1/365=0.0028=0.28%

b)If C=3

Probability that all 3 people have the same exact birthday=1-(364/365)*(364/365)=0.0055=0.55%

c)probability that some pair of the 2 people have the same exact birthday=1-(365/365)*(364/365)*(363/365)*=0.0082=0.82%

d)The probability that no one else has your birthday, in a crowd of size n,is

Qn=(364/365)n-1

In order for the probability of at least one other person to share your birthday to exceed 50%, we need n large enough that

1-Qn>=0.5

n>253

Many people find this surprising (and would expect the answer to be something like 365/2 183). One partial explanation for the counter-intuitive high answer is that, among the others, there are likely to be many pairs that share the same birthday; in 253 people, the number of distinct birthdays represented may be many fewer than 253.

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