Let \( E \) and \( F \) be two events such that \( P(E) = 0.1 \), \( P(F) = 0.3 \), \( P(E \cup F) = 0.4 \). Then \( P(F|E) \) is:
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- \( 0.6 \)
- \( 0.4 \)
- \( 0.5 \)
- \( 0 \)
The Correct Option is D
Solution and Explanation
Step 1: Recall the formula for conditional probability
The conditional probability \( P(F|E) \) is given by: \[ P(F|E) = \frac{P(E \cap F)}{P(E)}. \] Step 2: Find \( P(E \cap F) \)
Using the formula for the probability of the union of two events: \[ P(E \cup F) = P(E) + P(F) - P(E \cap F). \] Substitute the given values \( P(E \cup F) = 0.4 \), \( P(E) = 0.1 \), \( P(F) = 0.3 \): \[ 0.4 = 0.1 + 0.3 - P(E \cap F). \] Simplify to find \( P(E \cap F) \): \[ P(E \cap F) = 0.1 + 0.3 - 0.4 = 0. \] Step 3: Calculate \( P(F|E) \)
Substitute \( P(E \cap F) = 0 \) and \( P(E) = 0.1 \) into the formula for \( P(F|E) \): \[ P(F|E) = \frac{P(E \cap F)}{P(E)} = \frac{0}{0.1} = 0. \] Step 4: Conclude the result
The conditional probability \( P(F|E) \) is \( 0 \). This indicates that the events \( E \) and \( F \) do not overlap.
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