Question:
If \(E\) and \(F\) are two independent events such that \[ P(E)=\frac{3}{10}, \qquad P(E\cup F)=\frac{1}{2}, \] then \[ P(E|F)-P(F|E) \] is equal to:
If \(E\) and \(F\) are two independent events such that \[ P(E)=\frac{3}{10}, \qquad P(E\cup F)=\frac{1}{2}, \] then \[ P(E|F)-P(F|E) \] is equal to:
Show Hint
"Independent" means information about one event does not change the probability of the other.
So, \(P(A|B)\) is always just \(P(A)\).
This simplifies these problems to finding individual probabilities and performing basic arithmetic.
Updated On: Sep 11, 2026
- \(\frac{2}{7}\)
- \(\frac{3}{35}\)
- \(\frac{1}{70}\)
- \(\frac{1}{7}\)
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The Correct Option is C
Solution and Explanation
Concept:
• For independent events \(E\) and \(F\): \(P(E \cap F) = P(E) \cdot P(F)\).
• Conditional probability for independent events: \(P(E|F) = P(E)\) and \(P(F|E) = P(F)\).
• Addition Theorem: \(P(E \cup F) = P(E) + P(F) - P(E \cap F)\).
Step 1: Calculate \(P(F)\)
Using independence in the addition theorem: \[ P(E \cup F) = P(E) + P(F) - P(E)P(F) \] \[ \frac{1}{2} = \frac{3}{10} + P(F) - \frac{3}{10}P(F) \] \[ \frac{1}{2} - \frac{3}{10} = \frac{7}{10}P(F) \] \[ \frac{5 - 3}{10} = \frac{7}{10}P(F) \implies \frac{2}{10} = \frac{7}{10}P(F) \] \[ P(F) = \frac{2}{7} \]
Step 2: Evaluate conditional probabilities
Since the events are independent: \[ P(E|F) = P(E) = \frac{3}{10} \] \[ P(F|E) = P(F) = \frac{2}{7} \]
Step 3: Find the difference
\[ P(E|F) - P(F|E) = \frac{3}{10} - \frac{2}{7} \] To subtract, use a common denominator of \(70\): \[ \frac{3 \times 7 - 2 \times 10}{70} = \frac{21 - 20}{70} = \frac{1}{70} \]
• For independent events \(E\) and \(F\): \(P(E \cap F) = P(E) \cdot P(F)\).
• Conditional probability for independent events: \(P(E|F) = P(E)\) and \(P(F|E) = P(F)\).
• Addition Theorem: \(P(E \cup F) = P(E) + P(F) - P(E \cap F)\).
Step 1: Calculate \(P(F)\)
Using independence in the addition theorem: \[ P(E \cup F) = P(E) + P(F) - P(E)P(F) \] \[ \frac{1}{2} = \frac{3}{10} + P(F) - \frac{3}{10}P(F) \] \[ \frac{1}{2} - \frac{3}{10} = \frac{7}{10}P(F) \] \[ \frac{5 - 3}{10} = \frac{7}{10}P(F) \implies \frac{2}{10} = \frac{7}{10}P(F) \] \[ P(F) = \frac{2}{7} \]
Step 2: Evaluate conditional probabilities
Since the events are independent: \[ P(E|F) = P(E) = \frac{3}{10} \] \[ P(F|E) = P(F) = \frac{2}{7} \]
Step 3: Find the difference
\[ P(E|F) - P(F|E) = \frac{3}{10} - \frac{2}{7} \] To subtract, use a common denominator of \(70\): \[ \frac{3 \times 7 - 2 \times 10}{70} = \frac{21 - 20}{70} = \frac{1}{70} \]
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