Question

Given k people in a room, what is the probability that there are two who share a birthday? You can ignore leap years and assume all birthdays are equally likely. Hint: You can start thinking of the complementary event. (b) What is the lowest k for which the above probability exceeds 50%?

Answer 1

a)

here for above number of ways that all of k people have different birthday =N(number of ways to choose k days out of 365 without replacement) =³⁶⁵P_k

totoal ways to assign birthdays to k people =N(for each person we have 365 choices)=(365)^k

hence number of ways none of k have same birthday =³⁶⁵P_k/(365)^k

therefore probability that there are at least two who share a birthday =1-P(none share)

=1-³⁶⁵P_k/(365)^k=1-365*364*363*...(365-k+1)/(365)^k

b)

for above to exceed 50% ;

1-365*364*363*...(365-k+1)/(365)^k >=0.50

by hit and trail ;

k>=23

hence lowest k=23