To determine which of the given tables represents a valid probability distribution, we need to check two key criteria for a probability distribution:
1. All probabilities must be between 0 and 1 (inclusive).
2. The sum of all probabilities must equal 1.
Let's check each distribution step-by-step.
### Probability Distribution A
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution A } \\
\hline$X$ & $P(x)$ \\
\hline 1 & -0.14 \\
\hline 2 & 0.6 \\
\hline 3 & 0.25 \\
\hline 4 & 0.29 \\
\hline
\end{tabular}
\][/tex]
1. Check if each probability is between 0 and 1:
- [tex]\(P(1) = -0.14\)[/tex] (Not between 0 and 1)
- [tex]\(P(2) = 0.6\)[/tex]
- [tex]\(P(3) = 0.25\)[/tex]
- [tex]\(P(4) = 0.29\)[/tex]
Since [tex]\(-0.14\)[/tex] is not between 0 and 1, Probability Distribution A does not satisfy the first criterion.
### Probability Distribution B
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution B } \\
\hline$X$ & $P(x)$ \\
\hline 1 & 0 \\
\hline 2 & 0.45 \\
\hline 3 & 0.16 \\
\hline 4 & 0.39 \\
\hline
\end{tabular}
\][/tex]
1. Check if each probability is between 0 and 1:
- [tex]\(P(1) = 0\)[/tex]
- [tex]\(P(2) = 0.45\)[/tex]
- [tex]\(P(3) = 0.16\)[/tex]
- [tex]\(P(4) = 0.39\)[/tex]
All probabilities are between 0 and 1.
2. Sum of probabilities:
[tex]\[
P(1) + P(2) + P(3) + P(4) = 0 + 0.45 + 0.16 + 0.39 = 1
\][/tex]
The sum of probabilities is 1, so Probability Distribution B satisfies both criteria.
### Probability Distribution C
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{ Probability Distribution C } \\
\hline$X$ & $P(x)$ \\
\hline 1 & 0.45 \\
\hline 2 & 1.23 \\
\hline 3 & -0.87 \\
\hline
\end{tabular}
\][/tex]
1. Check if each probability is between 0 and 1:
- [tex]\(P(1) = 0.45\)[/tex]
- [tex]\(P(2) = 1.23\)[/tex] (Not between 0 and 1)
- [tex]\(P(3) = -0.87\)[/tex] (Not between 0 and 1)
Since 1.23 and -0.87 are not between 0 and 1, Probability Distribution C does not satisfy the first criterion.
Based on the analysis, only Probability Distribution B satisfies both criteria required for a valid probability distribution.
So, the correct and valid probability distribution is:
Probability Distribution B.
