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A family has two children. If [tex]\( B \)[/tex] represents a boy and [tex]\( G \)[/tex] represents a girl, the set of outcomes for the possible genders of the children is [tex]\( S = \{BB, BG, GB, GG\} \)[/tex], with the oldest child listed first in each pair. Let [tex]\( X \)[/tex] represent the number of times [tex]\( G \)[/tex] occurs. Which of the following is the probability distribution, [tex]\( P_X(x) \)[/tex]?

\begin{tabular}{|c|c|}
\hline
[tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline
0 & 0.25 \\
\hline
1 & 0.5 \\
\hline
2 & 0.25 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline
[tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline
0 & 0.33 \\
\hline
1 & 0.33 \\
\hline
2 & 0.33 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline
[tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline
0 & 0.25 \\
\hline
1 & 0.75 \\
\hline
2 & 0 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline
[tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline
0 & 0 \\
\hline
1 & 0 \\
\hline
2 & 0 \\
\hline
\end{tabular}


Sagot :

To determine the probability distribution of [tex]\( X \)[/tex], the number of times [tex]\( G \)[/tex] (girl) occurs, we start by understanding the sample space of outcomes for the genders of the two children, given by:

[tex]\[ S = \{BB, BG, GB, GG\} \][/tex]

There are four equally likely outcomes.

Now, let's identify the probabilities for each value of [tex]\( X \)[/tex]:

1. [tex]\( X = 0 \)[/tex]: This corresponds to the outcome where there are no girls, which is [tex]\( BB \)[/tex].

- There is 1 outcome in the sample space that fits this description.
- So, the probability [tex]\( P(X = 0) \)[/tex] is calculated as:
[tex]\[ P(X = 0) = \frac{\text{Number of outcomes with no girl}}{\text{Total number of outcomes}} = \frac{1}{4} = 0.25 \][/tex]

2. [tex]\( X = 1 \)[/tex]: This corresponds to the outcomes where there is exactly one girl, which are [tex]\( BG \)[/tex] and [tex]\( GB \)[/tex].

- There are 2 outcomes that fit this description.
- So, the probability [tex]\( P(X = 1) \)[/tex] is calculated as:
[tex]\[ P(X = 1) = \frac{\text{Number of outcomes with exactly one girl}}{\text{Total number of outcomes}} = \frac{2}{4} = 0.5 \][/tex]

3. [tex]\( X = 2 \)[/tex]: This corresponds to the outcome where there are two girls, which is [tex]\( GG \)[/tex].

- There is 1 outcome that fits this description.
- So, the probability [tex]\( P(X = 2) \)[/tex] is calculated as:
[tex]\[ P(X = 2) = \frac{\text{Number of outcomes with two girls}}{\text{Total number of outcomes}} = \frac{1}{4} = 0.25 \][/tex]

Summarizing these probabilities:

[tex]\[ P_X(x) = \begin{cases} 0.25 & \text{if } x = 0 \\ 0.5 & \text{if } x = 1 \\ 0.25 & \text{if } x = 2 \\ \end{cases} \][/tex]

Therefore, the correct probability distribution for [tex]\( X \)[/tex] is:

\begin{tabular}{|c|c|}
\hline
[tex]\( X \)[/tex] & [tex]\( P(X) \)[/tex] \\
\hline
0 & 0.25 \\
\hline
1 & 0.5 \\
\hline
2 & 0.25 \\
\hline
\end{tabular}

This matches the first given table, confirming it is the correct probability distribution.