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Suppose [tex]A[/tex] and [tex]B[/tex] are dependent events. If [tex]P(A)=0.4[/tex] and [tex]P(B \mid A)=0.8[/tex], what is [tex]P(A \cap B)[/tex]?

A. 0.2
B. 0.32
C. 0.4
D. 0.8

Sagot :

GinnyAnswer
Certainly! Let's solve this step by step.

Given:
- \( P(A) = 0.4 \)
- \( P(B \mid A) = 0.8 \)

We need to find \( P(A \cap B) \), which is the probability that both events \( A \) and \( B \) occur simultaneously.

From the definition of conditional probability, we know that:
[tex]\[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \][/tex]

We can rearrange this formula to solve for \( P(A \cap B) \):
[tex]\[ P(A \cap B) = P(B \mid A) \times P(A) \][/tex]

Now, substitute the given values:
[tex]\[ P(A \cap B) = 0.8 \times 0.4 \][/tex]

Multiplying the values, we get:
[tex]\[ P(A \cap B) = 0.32 \][/tex]

Thus, the probability \( P(A \cap B) \) is \( 0.32 \).

The correct answer is:
B. 0.32
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