Question

A family has three children. If the genders of these children are listed in the order they are born, there are eight possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely. Let X represent the number of children that are girls. Find the probability distribution of X.

Answer 1

To find the probability distribution of \( X \), which represents the number of girls in a family with three children, we first list all the possible outcomes and count the number of girls in each outcome:

• Outcome: BBB (0 girls)
• Outcome: BBG (1 girl)
• Outcome: BGB (1 girl)
• Outcome: BGG (2 girls)
• Outcome: GBB (1 girl)
• Outcome: GBG (2 girls)
• Outcome: GGB (2 girls)
• Outcome: GGG (3 girls)

Next, we can summarize the number of girls (X) associated with each outcome:

• \( X = 0 \): 1 outcome (BBB)
• \( X = 1 \): 3 outcomes (BBG, BGB, GBB)
• \( X = 2 \): 3 outcomes (BGG, GBG, GGB)
• \( X = 3 \): 1 outcome (GGG)

Now, we count the total outcomes for each value of \( X \):

• \( P(X = 0) = \frac{1}{8} \) (1 outcome)
• \( P(X = 1) = \frac{3}{8} \) (3 outcomes)
• \( P(X = 2) = \frac{3}{8} \) (3 outcomes)
• \( P(X = 3) = \frac{1}{8} \) (1 outcome)

Now we can summarize the probability distribution of \( X \):

\[ \begin{align*} P(X = 0) & = \frac{1}{8} \ P(X = 1) & = \frac{3}{8} \ P(X = 2) & = \frac{3}{8} \ P(X = 3) & = \frac{1}{8} \ \end{align*} \]

Lastly, we can explicitly list the full distribution:

• \( P(X) \):
  • \( P(X = 0) = \frac{1}{8} \)
  • \( P(X = 1) = \frac{3}{8} \)
  • \( P(X = 2) = \frac{3}{8} \)
  • \( P(X = 3) = \frac{1}{8} \)

This is the probability distribution of \( X \).

Answer 2

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

Answer 3

The random variable \( X \) represents the number of girls in a family with three children. Since the number of girls can range from 0 (when all children are boys) to 3 (when all children are girls), the possible values of \( X \) are:

• \( X = 0 \) (no girls)
• \( X = 1 \) (one girl)
• \( X = 2 \) (two girls)
• \( X = 3 \) (three girls)

Thus, the possible values for the random variable \( X \) are:

• \( 0, 1, 2, 3 \)

To find the number of possible values, we simply count these:

• There are 4 possible values for \( X \).

Answer: The number of possible values for the random variable \( X \) is 4.

Answer 4

(b) Find the probability distribution of X. Round your answers to four decimals as needed.
x| 0 1 2 3
P(x)|

Answer 5

We can summarize the probability distribution of \( X \), the number of girls in a family with three children, based on the outcomes we identified earlier.

The probabilities for each value \( X \) are calculated based on the possible outcomes of having 0, 1, 2, or 3 girls. The counts from before were:

• For \( X = 0 \): 1 outcome (BBB)
• For \( X = 1 \): 3 outcomes (BBG, BGB, GBB)
• For \( X = 2 \): 3 outcomes (BGG, GBG, GGB)
• For \( X = 3 \): 1 outcome (GGG)

Since there are a total of 8 equally likely outcomes, the probabilities can be calculated as follows:

\[ \begin{align*} P(X = 0) & = \frac{1}{8} = 0.1250 \ P(X = 1) & = \frac{3}{8} = 0.3750 \ P(X = 2) & = \frac{3}{8} = 0.3750 \ P(X = 3) & = \frac{1}{8} = 0.1250 \ \end{align*} \]

Rounding to four decimals as specified, the final probability distribution of \( X \) is:

\[ \begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \ \hline P(X) & 0.1250 & 0.3750 & 0.3750 & 0.1250 \ \end{array} \]

So the final answer is:

\[ \begin{array}{c|cccc} x & 0 & 1 & 2 & 3 \ \hline P(x) & 0.1250 & 0.3750 & 0.3750 & 0.1250 \ \end{array} \]