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, $p_x(x)$?
four tables with x and $p_x(x)$ values are shown:

1. x: 0,1,2; $p_x(x)$: 0.33,0.33,0.33
2. x: 0,1,2; $p_x(x)$: 0.25,0.75,0
3. x: 0,1,2; $p_x(x)$: 0.25,0.5,0.25
4. x: 0,1,2; $p_x(x)$: 0,0.5,0.5

Explanation:

Step1: Determine possible values of \( X \)

The number of girls \( X \) can be 0, 1, or 2. We analyze each outcome in \( S = \{BB, BG, GB, GG\} \):

• For \( BB \), \( X = 0 \) (no girls).
• For \( BG \) and \( GB \), \( X = 1 \) (one girl each).
• For \( GG \), \( X = 2 \) (two girls).

Step2: Calculate probabilities

• Probability \( P(X = 0) \): Only \( BB \) gives \( X = 0 \). There is 1 outcome out of 4, so \( P(X = 0) = \frac{1}{4} = 0.25 \).
• Probability \( P(X = 1) \): \( BG \) and \( GB \) give \( X = 1 \). There are 2 outcomes out of 4, so \( P(X = 1) = \frac{2}{4} = 0.5 \).
• Probability \( P(X = 2) \): Only \( GG \) gives \( X = 2 \). There is 1 outcome out of 4, so \( P(X = 2) = \frac{1}{4} = 0.25 \).

Step3: Match with given tables

We check which table has \( P(X = 0) = 0.25 \), \( P(X = 1) = 0.5 \), and \( P(X = 2) = 0.25 \). The third table (bottom - left) matches these probabilities.

Answer:

The probability distribution \( P_X(x) \) is the table with:

\( X \)	\( P_X(x) \)
1	0.5
2	0.25