Question

Suppose you pick people at random and ask them what month of the year they were born in. Let X be the number of people you have to ask until you findnd a person who was born in December. (Just assume each month is equally likely to make it simpler.)

A) Find the probability that you had to ask exactly 9 people given that you had to ask at least 3 people. Ans: 0.04944

Answer 1

Let p denote the probability that a randomly selected person is born in December.

Now, since there are 12 months and each person is equally likely to be born in any of these months, thus we get:

p = P(a person is born in December) = 1/12

=> 1 - p = P(a person is not born in December) = 11/12

Now, we are given that X is the number of people we have to ask until we find a person who was born in December.

Now, the first person we ask could be born in December (X=1), the second person we ask could be the first person who was born in December (X=2), the third person we ask could be the first person who was born in December (X=3) and so on. Thus, the possible value of X are 1,2,3,...

Now, we find the probability mass function of X:

P(X=x) = P(x^th person is the first person who was born in December)

= P(the first (x-1) persons were not born in December and the x^th person was born in December)

= P(first person was not born in December)*P(second person was not born in December)*...*P((x-1)^th person was not born in December)*(xth person was born in December)

[Since, the persons are independent]

Now, the probability that you had to ask exactly 9 people given that you had to ask at least 3 people is given by:

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