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Find [tex]$P(C \mid Y)$[/tex] from the information in the table.

\begin{tabular}{|c|c|c|c|c|}
\hline
& [tex]$X$[/tex] & [tex]$Y$[/tex] & [tex]$Z$[/tex] & Total \\
\hline
A & 32 & 10 & 28 & 70 \\
\hline
B & 6 & 5 & 25 & 36 \\
\hline
C & 18 & 15 & 7 & 40 \\
\hline
Total & 56 & 30 & 60 & 146 \\
\hline
\end{tabular}

To the nearest tenth, what is the value of [tex]$P(C \mid Y)$[/tex]?

A. 0.4
B. 0.5
C. 0.7
D. 0.8

Sagot :

GinnyAnswer
To find [tex]\( P(C \mid Y) \)[/tex], you need to determine the probability of event [tex]\( C \)[/tex] occurring given that event [tex]\( Y \)[/tex] has occurred. This is calculated using conditional probability, which is defined as:

[tex]\[ P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)} \][/tex]

where:
- [tex]\( P(C \cap Y) \)[/tex] is the joint probability of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occurring.
- [tex]\( P(Y) \)[/tex] is the probability of [tex]\( Y \)[/tex] occurring.

From the given table, let's identify these values step-by-step:

1. Identify the joint occurrence of [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] (i.e., the value corresponding to [tex]\( C \)[/tex] in row [tex]\( C \)[/tex] and column [tex]\( Y \)[/tex]):
- [tex]\( P(C \cap Y) \)[/tex] is represented by the value 15.

2. Identify the total occurrences of [tex]\( Y \)[/tex] (i.e., the total value in column [tex]\( Y \)[/tex]):
- [tex]\( P(Y) \)[/tex] is represented by the total value 30.

Using these values, calculate [tex]\( P(C \mid Y) \)[/tex]:

[tex]\[ P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)} = \frac{15}{30} = 0.5 \][/tex]

Thus, the value of [tex]\( P(C \mid Y) \)[/tex] to the nearest tenth is:

[tex]\[ \boxed{0.5} \][/tex]
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