Question

If \( P(A)=0.4 \) and \( P(B|A)=0.9 \), then \( P(A \cap B) \) is equal to

• A. \( 0.36 \)
• B. \( 0.6 \)
• C. \( 0.64 \)
• D. \( 0.44 \)
• E. \( 0.24 \)

Hint:

Always remember the key formula \( P(A \cap B)=P(B|A)\cdot P(A) \). It is the fastest way to solve conditional probability problems.

Answer 1

The Correct Option is A

Step 1: Recall the definition of conditional probability.
The conditional probability of event \( B \) given event \( A \) is defined as: \[ P(B|A)=\frac{P(A \cap B)}{P(A)} \]

Step 2: Rearrange the formula.
From the above formula, we can write: \[ P(A \cap B)=P(B|A)\cdot P(A) \]

Step 3: Substitute the given values.
We are given: \[ P(A)=0.4,\qquad P(B|A)=0.9 \] Substitute into the formula: \[ P(A \cap B)=0.9 \times 0.4 \]

Step 4: Perform the multiplication.
\[ 0.9 \times 0.4=0.36 \]

Step 5: Interpret the result.
This means the probability that both events \( A \) and \( B \) occur together is \( 0.36 \).

Step 6: Check the validity.
Since probabilities must lie between \( 0 \) and \( 1 \), the value \( 0.36 \) is valid.

Step 7: Final conclusion.
Therefore, \[ \boxed{P(A \cap B)=0.36} \] Hence, the correct option is \[ \boxed{(1)\ 0.36} \]