Question

If $P(A)=0.7, P(B)=0.5$ and $P(A\cup B)=0.9$, then $P(A/B)$ is ________.

• A. 0.3
• B. 0.4
• C. 0.5
• D. 0.6
• E. 0.7

Hint:

Always find the intersection first for conditional probability problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
This problem involves conditional probability. We need to find the probability of event A occurring given that event B has already occurred. This requires us to first find the probability of the intersection of A and B.
Step 2: Key Formula or Approach
We will use two fundamental formulas of probability:
1. Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). We will use this to find \(P(A \cap B)\).
2. Conditional Probability Formula: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).
Step 3: Detailed Explanation
1. Find \(P(A \cap B)\) using the Addition Rule.
We are given \(P(A) = 0.7\), \(P(B) = 0.5\), and \(P(A \cup B) = 0.9\).
Rearranging the addition rule to solve for the intersection:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \] Substitute the given values:
\[ P(A \cap B) = 0.7 + 0.5 - 0.9 \] \[ P(A \cap B) = 1.2 - 0.9 = 0.3 \] 2. Calculate \(P(A|B)\) using the Conditional Probability Formula.
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \] Substitute the value of \(P(A \cap B)\) we just found and the given value of \(P(B)\):
\[ P(A|B) = \frac{0.3}{0.5} \] \[ P(A|B) = \frac{3}{5} = 0.6 \] Step 4: Final Answer
The value of \(P(A|B)\) is 0.6.