A family has two children. If the genders of these children are listed in the order they are born, there are four possible outcomes: BB, BG, GB, and GG. Assume these outcomes are equally likely. Let X represent the number of children that are girls. Find the probability distribution of X.

Question

(a) Find the number of possible values for the random variable X

Answer 1

The random variable \( X \) represents the number of girls among the two children. The possible outcomes for the genders of the two children are:

• BB (0 girls)
• BG (1 girl)
• GB (1 girl)
• GG (2 girls)

Now, we can summarize the possible values of \( X \):

• If both children are boys (BB), then \( X = 0 \).
• If one child is a boy and the other is a girl (BG or GB), then \( X = 1 \).
• If both children are girls (GG), then \( X = 2 \).

Thus, the possible values for \( X \) are \( 0, 1, \) and \( 2 \).

(a) The number of possible values for the random variable \( X \) is 3 (i.e., \( X \) can take on the values 0, 1, or 2).

Answer 2

(b) Find the probability distribution ofX . Round your answers to four decimals as needed.
X| 0 1 2
P(x)| [?] [?] [?]

Answer 3

To find the probability distribution of \( X \), we first determine the probabilities for each possible value of \( X \):

1. \( X = 0 \): This occurs when both children are boys (BB). The outcome is:

   • \( \text{Outcome: BB} \)

   Probability: \[ P(X = 0) = \frac{1}{4} \]

2. \( X = 1 \): This occurs when one child is a boy and the other is a girl (BG or GB). The outcomes are:

   • \( \text{Outcomes: BG, GB} \)

   Probability: \[ P(X = 1) = \frac{2}{4} = \frac{1}{2} \]

3. \( X = 2 \): This occurs when both children are girls (GG). The outcome is:

   • \( \text{Outcome: GG} \)

   Probability: \[ P(X = 2) = \frac{1}{4} \]

Now, we can summarize the probability distribution in table form, rounding to four decimal places where needed:

| \( X \) | Probability \( P(X) \) | |---------|-----------------------| | 0 | 0.2500 | | 1 | 0.5000 | | 2 | 0.2500 |

Thus, the probability distribution of \( X \) is:

• \( P(X = 0) = 0.2500 \)
• \( P(X = 1) = 0.5000 \)
• \( P(X = 2) = 0.2500 \)