Question

A couple plans to have children until they get a boy, but they agree that they will not have more than four children even if all are girls. Find the expected number of children they will have. Assume that boys and girls are equally likely. (Round your answer to three decimal places.)

Answer 1

Answer:

1.875  is the expected number of children the couple will have.

Explanation:

The probability of an event X is denoted by P(X).

Let assume that X is the number of children to whom the couple will give birth. The couple can have at most four children with the following probabilities.

• The couple will have one child if the first is boy, thus:

For one child P(X)=1/2=0.50

• The couple will have two children if the first is girl and the second is a boy, thus:

For two children P(X)=0.5×0.5=0.25

• The couple will have three children if the first two are girls and the third is a boy, thus:

For three children P(X)= 0.5×0.5×0.5=0.125

• The couple will have four children if the first three are girls and the fourth is a boy, or if all the four are girls. thus:

For four children P(X) =(0.5×0.5×0.5×0.5)+(0.5×0.5×0.5×0.5)=0.125

Now calculating the expected number of  children the couple will have:

NOTE: Formula is given in the attached pic

Expected number of children = E(X)= (1×0.50)+(2×0.25)+(3×0.125)+(4×0.125)

E(X)= 1.875