Question

In: Statistics and Probability

Suppose you pick people at random and ask them what month of the year they were...

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

Solutions

Expert Solution

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(xth person is the first person who was born in December)

= P(the first (x-1) persons were not born in December and the xth 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:

For any queries, feel free to comment and ask.

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