Question

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.

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

Answer 1

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

Answer 2

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 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:

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