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]
### 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]