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Which of the following is a valid probability distribution?

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Probability Distribution A} \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & 0.42 \\
\hline 2 & 0.38 \\
\hline 3 & 0.13 \\
\hline 4 & 0.07 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Probability Distribution B} \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & 0.27 \\
\hline 2 & 0.28 \\
\hline 3 & 0.26 \\
\hline 4 & 0.27 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline Probability Distribution C \\
\hline [tex]$X$[/tex] & [tex]$P(x)$[/tex] \\
\hline 1 & 0.16 \\
\hline 2 & 0.39 \\
\hline 3 & 0.18 \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
To determine which of the given probability distributions are valid, we need to verify if the sum of probabilities in each distribution equals 1. A valid probability distribution must have probabilities that sum to exactly 1.

Let's analyze each distribution in detail:

### Probability Distribution A:

Given probabilities:
[tex]\[P(X = 1) = 0.42\][/tex]
[tex]\[P(X = 2) = 0.38\][/tex]
[tex]\[P(X = 3) = 0.13\][/tex]
[tex]\[P(X = 4) = 0.07\][/tex]

Sum of probabilities:
[tex]\[0.42 + 0.38 + 0.13 + 0.07 = 1.0\][/tex]

Since the sum is 1.0, Distribution A is a valid probability distribution.

### Probability Distribution B:

Given probabilities:
[tex]\[P(X = 1) = 0.27\][/tex]
[tex]\[P(X = 2) = 0.28\][/tex]
[tex]\[P(X = 3) = 0.26\][/tex]
[tex]\[P(X = 4) = 0.27\][/tex]

Sum of probabilities:
[tex]\[0.27 + 0.28 + 0.26 + 0.27 = 1.08\][/tex]

Since the sum is 1.08, which is not equal to 1, Distribution B is not a valid probability distribution.

### Probability Distribution C:

Given probabilities:
[tex]\[P(X = 1) = 0.16\][/tex]
[tex]\[P(X = 2) = 0.39\][/tex]
[tex]\[P(X = 3) = 0.18\][/tex]

Sum of probabilities:
[tex]\[0.16 + 0.39 + 0.18 = 0.73\][/tex]

Since the sum is 0.73, which is not equal to 1, Distribution C is not a valid probability distribution.

### Summary:

- Probability Distribution A: Valid (sum = 1.0)
- Probability Distribution B: Not Valid (sum = 1.08)
- Probability Distribution C: Not Valid (sum = 0.73)

Thus, the only valid probability distribution among the given options is Probability Distribution A.
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