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Chapter 11: Problem 21

Explain how to find the expected value for the number of girls for a family with two children. What is the expected value?

Short Answer

Expert verified
The expected value for the number of girls in a family with two children is 1.

Step by step solution

01

List all possible outcomes

The possible outcomes of the genders of two children are: BB, BG, GB, GG. In this case, B represents a boy and G represents a girl. So, we can have either 0, 1, or 2 girls.
02

Calculate the probability for each event

With two children, the number of girls can be 0 (BB), 1 (BG or GB), or 2 (GG). The probability of having 0 girl (BB) is \(0.5 * 0.5 = 0.25\). The probability of having 1 girl (BG or GB) is \(0.5 * 0.5 + 0.5 * 0.5 = 0.5\). The probability of having 2 girls (GG) is \(0.5 * 0.5 = 0.25\).
03

Determine the expected value

The expected value is calculated using the formula \(E(X) = \sum [x_i * P(x_i)]\), where \(x_i\) represents the possible values of girls (0, 1, 2) and P(xi) represents their corresponding probabilities. Thus, \(E(X) = 0 * 0.25 + 1 * 0.5 + 2 * 0.25 = 1\).

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