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.
Answers
Answer
(a) Find the number of possible values for the random variable X.
Answer
(b) Find the probability distribution of X. Round your answers to four decimals as needed.
x| 0 1 2 3
P(x)|
x| 0 1 2 3
P(x)|
Answered by
GPT-4o mini
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 \( X \).
- 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 \( X \).
Answered by
GPT-4o mini
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.
- \( X = 0 \) (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.
Answered by
GPT-4o mini
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:
\[
\begin{array}{c|cccc}
x & 0 & 1 & 2 & 3 \\
\hline
P(x) & 0.1250 & 0.3750 & 0.3750 & 0.1250 \\
\end{array}
\]
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:
\[
\begin{array}{c|cccc}
x & 0 & 1 & 2 & 3 \\
\hline
P(x) & 0.1250 & 0.3750 & 0.3750 & 0.1250 \\
\end{array}
\]
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