Question

a family has two children. if b represents a boy and g represents a girl, the set of outcomes for the possible genders of the children is ( s={bb,bg,gb,gg} ), with the oldest child listed first in each pair. let x represent the number of times g occurs. which of the following is the probability distribution, ( p_x(x) )?

Explanation:

Step1: Calculate probability for \(X = 0\)

When \(X = 0\) (no girls, i.e., \(BB\)), number of favorable outcomes \(n=1\). Total outcomes \(N = 4\). Probability \(P(X = 0)=\frac{1}{4}=0.25\)

Step2: Calculate probability for \(X = 1\)

When \(X = 1\) (one girl: \(BG\) or \(GB\)), number of favorable outcomes \(n = 2\). Probability \(P(X = 1)=\frac{2}{4}=0.5\)

Step3: Calculate probability for \(X = 2\)

When \(X = 2\) (two girls: \(GG\)), number of favorable outcomes \(n=1\). Probability \(P(X = 2)=\frac{1}{4}=0.25\)

Answer:

The first table (where \(X = 0\), \(P_X(x)=0.25\); \(X = 1\), \(P_X(x)=0.5\); \(X = 2\), \(P_X(x)=0.25\)) is the correct probability distribution.