1. The probability that a child being born will be a girl is 50%. Suppose couple

ID: 2929380 • Letter: 1

Question

1. The probability that a child being born will be a girl is 50%. Suppose couple really wants to have a girl.

(a) What is the expected number of children until they get a daughter?

(b) What is the probability that their rst daughter will be the fourth child?

(c) What is the probability that you will have to have at least six children but fewer than ten in order to get their rst daughter?

(d) Suppose that the couple already had four sons. What is the probability that their rst daughter will be their seventh child?

(a) p = 0.5

n = 1

=> Expected value = np = 0.5*1 = 0.5.

(b) If the first daughter is the fourth child, the first three children must be boys.

Probability that the first three children are boys = 0.5 * 0.5 * 0.5 = 0.125

=> Probability that the first daughter is the fourth child = 0.125 * 0.5 = 0.0625.

Probability that there are no girls in the next four children = 0.54.

=> Probability that there is atleast one girl in the next four children = 1 - 0.54.

=> Probability that there is at least six children but fewer than ten in order to get their rst daughter = 0.55 * (1 - 0.54) = 0.02930.

(d) Since the couple had four sons, the only way the first daughter is the seventh child is if the next two are boys and the third is a girl.

Probability that the next two are boys and the third is a girl = 0.5 * 0.5 * 0.5 = 0.125.

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