Question

If \(P(B) = \frac{9}{13}\) and \(P(A \cap B) = \frac{4}{13}\), find \(P(A|B)\).

Hint:

The notation \(P(A|B)\) can be read as "the probability of A, given B". The event that is "given" always goes in the denominator of the conditional probability formula. It represents the reduced sample space.

Answer 1

Step 1: Understanding the Concept:
This question asks for the conditional probability of event A given that event B has occurred, denoted as \(P(A|B)\). This is a direct application of the definition of conditional probability.
Step 2: Key Formula or Approach:
The formula for the conditional probability of A given B is: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] where \(P(A \cap B)\) is the probability of both A and B occurring, and \(P(B)\) is the probability of B occurring, with the condition that \(P(B) \neq 0\).
Step 3: Detailed Explanation or Calculation:
We are given the following probabilities: \[ P(B) = \frac{9}{13} \] \[ P(A \cap B) = \frac{4}{13} \] Using the formula for conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{4/13}{9/13} \] The denominators (13) cancel out: \[ P(A|B) = \frac{4}{9} \] Step 4: Final Answer:
The value of \(P(A|B)\) is \(\frac{4}{9}\).