Question

Assume that there are 365 days in a year, the probability that a person is born on any given day is the same for all days, and everyone’s birthdays are independent.

(a) If there are n people, what is the probability that no one is born in July?

(b) If there are 15 people who are all born in July or August, what is the probability that at least two of these people share the same birthday?

For part a, wouldn't it just be 365^n - 365^31 since there are only 31 days in July? Is there a way to make it more order specific? For b I am still confused on how to make order matter if we are just taking days from July or August. Explanations well appreciated!

Answer 1

Explanation:

If each day equal chance then p = Prob that a person is borne on a particular day = 1/365

Each person is independent of the other and there are two outcomes either borne in July or not

p = prob for one person not borne in July = (365-31)/365 = 334/365

a)Hence prob that no one from n people borne in July =
`((334)/(365) )^n`

b) p = prob of any one borne in July or Aug =
`((31+31) )/(365) =(62)/(365)`=0.1698

X- no of people borne in July or Aug

n =15

P(X>=2) =
`15C2 (0.1698)^2*(1-0.1698)^(13) +15C3 (0.1698)^3*(1-0.1698)^(12) +...+15C15 (0.1698)^(15)`

=0.7505