Question

1. the family many couples want to have both a boy and a girl. if they decide to continue to have children until they have one child of each sex, what would the average family size be? assume that boys and girls are equally likely.

Explanation:

Step1: Define the probability of having a boy or a girl

The probability of having a boy \(P(B)=\frac{1}{2}\), and the probability of having a girl \(P(G)=\frac{1}{2}\).

Step2: Calculate the expected value

Let \(X\) be the family size.

• If the first - born is a boy (\(P(B)=\frac{1}{2}\)), then we need to keep having children until we get a girl. The expected number of additional children after the first - born boy is \(E(X|B) = 1+\sum_{n = 1}^{\infty}n\times(\frac{1}{2})^{n}\). Using the formula for the sum of an infinite series \(\sum_{n = 1}^{\infty}nx^{n}=\frac{x}{(1 - x)^{2}}\) (where \(x=\frac{1}{2}\)), we have \(E(X|B)=1 + 2=3\).
• If the first - born is a girl (\(P(G)=\frac{1}{2}\)), then we need to keep having children until we get a boy. By symmetry, \(E(X|G)=3\)

Using the law of total expectation \(E(X)=P(B)\times E(X|B)+P(G)\times E(X|G)\)

Substitute \(P(B) = P(G)=\frac{1}{2}\), \(E(X|B)=E(X|G) = 3\) into the formula:

\(E(X)=\frac{1}{2}\times3+\frac{1}{2}\times3\)

\(E(X)= 3\)

Answer:

The average family size is \(3\)