Question

what is the expected number of female children in a family with three children? 2 1.5 2.5 1

Explanation:

Step1: Determine the probability distribution

The number of female children \(X\) can be \(0\), \(1\), \(2\), or \(3\). The probability of having \(k\) female children in \(n = 3\) children follows the binomial distribution \(P(X=k)={n\choose k}p^{k}(1 - p)^{n - k}\), where \(p=\frac{1}{2}\) (probability of a child being female).
\({3\choose0}(\frac{1}{2})^{0}(1-\frac{1}{2})^{3}=\frac{1}{8}\), \({3\choose1}(\frac{1}{2})^{1}(1-\frac{1}{2})^{2}=\frac{3}{8}\), \({3\choose2}(\frac{1}{2})^{2}(1-\frac{1}{2})^{1}=\frac{3}{8}\), \({3\choose3}(\frac{1}{2})^{3}(1-\frac{1}{2})^{0}=\frac{1}{8}\)

Step2: Calculate the expected value

The formula for the expected value \(E(X)=\sum_{k = 0}^{n}k\times P(X = k)\)

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Answer:

\(1.5\)