An number cube (a fair die) is rolled 3 times. For each roll, we are interested in whether the roll comes up even or odd. An outcome is represented by a string like "oee" (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll).

For each outcome, let [tex]$N$[/tex] be the random variable counting the number of even rolls in each outcome. For example, if the outcome is "eee," then [tex]$N(\text{eee}) = 3$[/tex]. Suppose that the random variable [tex]$X$[/tex] is defined in terms of [tex]$N$[/tex] as follows:
[tex]\[ X = 2N - 2N^2 - 3 \][/tex]

The values of [tex]$X$[/tex] are given in the table below.
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline Outcome & oee & eeo & ooe & oeo & eee & ooo & eoe & eoo \\
\hline Value of $X$ & -7 & -7 & -3 & -3 & -15 & -3 & -7 & -3 \\
\hline
\end{tabular}
\][/tex]

Calculate the probabilities [tex]$P(X=x)$[/tex] of the probability distribution of [tex]$X$[/tex]. First, fill in the first row with the values of [tex]$X$[/tex]. Then fill in the appropriate probabilities in the second row.
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline Value $x$ of $X$ & $\square$ & $\square$ & $\square$ \\
\hline $P(X=x)$ & $\square$ & $\square$ & $\square$ \\
\hline
\end{tabular}
\][/tex]



Answer :

Sure, let's solve the problem step by step to find the probability distribution of [tex]\( X \)[/tex].

### Step 1: Identify Unique Values of [tex]\( X \)[/tex]
Given these outcomes:

[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Outcome & oee & eeo & ooe & oeo & eee & ooo & eoe & eoo \\ \hline Value of \( X \) & -7 & -7 & -3 & -3 & -15 & -3 & -7 & -3 \\ \hline \end{tabular} \][/tex]

First, list out the unique values of [tex]\( X \)[/tex]:
[tex]\[ \{-7, -7, -3, -3, -15, -3, -7, -3\} \][/tex]

The unique values of [tex]\( X \)[/tex] from the list are:
[tex]\[ \{-7, -3, -15\} \][/tex]

### Step 2: Count Frequencies of Each Unique Value
Now, let's count the occurrences of each unique value of [tex]\( X \)[/tex]:
- [tex]\( -7 \)[/tex] appears 3 times
- [tex]\( -3 \)[/tex] appears 4 times
- [tex]\( -15 \)[/tex] appears 1 time

### Step 3: Calculate Probabilities
The total number of outcomes is 8. To find the probabilities, divide the frequency of each value by the total number of outcomes.

[tex]\[ P(X = -7) = \frac{3}{8} = 0.375 \][/tex]
[tex]\[ P(X = -3) = \frac{4}{8} = 0.5 \][/tex]
[tex]\[ P(X = -15) = \frac{1}{8} = 0.125 \][/tex]

### Step 4: Complete the Table
Now, fill in the table with the unique values of [tex]\( X \)[/tex] and their corresponding probabilities:

[tex]\[ \begin{tabular}{|c|c|c|c|} \hline Value \( x \) of \( X \) & -7 & -3 & -15 \\ \hline \( P(X = x) \) & 0.375 & 0.5 & 0.125 \\ \hline \end{tabular} \][/tex]

In summary, the probabilities [tex]\( P(X = x) \)[/tex] are correctly computed, and the filled-in table is as follows:

[tex]\[ \begin{tabular}{|c|c|c|c|} \hline Value \( x \) of \( X \) & -7 & -3 & -15 \\ \hline \( P(X = x) \) & 0.375 & 0.5 & 0.125 \\ \hline \end{tabular} \][/tex]