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) )? three tables with x and ( p_x(x) ) values (first: x=0→0.25, x=1→0.5, x=2→0.25; second: x=0→0.33, x=1→0.33, x=2→0.33; third: x=0→0.25, x=1→0.75, x=2→0) and radio buttons

Explanation:

Step1: Determine possible \( X \) values

The number of girls \( X \) can be 0, 1, or 2.

Step2: Calculate \( P(X = 0) \)

\( X = 0 \) means no girls, so the outcome is \( BB \). There is 1 such outcome out of 4 total (\( S=\{BB, BG, GB, GG\} \)). So \( P(X = 0)=\frac{1}{4}=0.25 \).

Step3: Calculate \( P(X = 1) \)

\( X = 1 \) means one girl. The outcomes are \( BG \) and \( GB \). There are 2 such outcomes. So \( P(X = 1)=\frac{2}{4}=0.5 \).

Step4: Calculate \( P(X = 2) \)

\( X = 2 \) means two girls, so the outcome is \( GG \). There is 1 such outcome. So \( P(X = 2)=\frac{1}{4}=0.25 \).

Answer:

The first table (with \( X = 0 \) having \( P_X(x)=0.25 \), \( X = 1 \) having \( P_X(x)=0.5 \), and \( X = 2 \) having \( P_X(x)=0.25 \))