Probability
Union and Intersection of Events
Grade 12
Question:
<p>Let A and B be two events such that the probability that exactly one of them occurs is <span>\(\frac{2}{5}\)</span> and the probability that A or B occurs is <span>\(\frac{1}{2}\)</span>, then the probability of both of them occur together is</p>
<p>(a) 0.10</p>
<p>(b) 0.20</p>
<p>(c) 0.01</p>
<p>(d) 0.02</p>
Step-by-Step Solution
Key Concept: Use the relation between probability of exactly one event occurring and probability of union of two events to find the probability of intersection.
<p><strong>Solution:</strong></p><p>For two events A and B it is given that probability of occurrence of exactly one of them is <span>$\frac{2}{5}$</span>.</p><p>We know that:</p><p>Probability of exactly one event occurs = <span>$P(A \cap B^c) + P(A^c \cap B) = P(A) + P(B) - 2P(A \cap B) = \frac{2}{5}$</span></p><p>Probability that A or B occurs = <span>$P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{1}{2}$</span></p><p>Let <span>$P(A \cap B) = x$</span></p><p>From the second equation: <span>$P(A) + P(B) = \frac{1}{2} + x$</span></p><p>Substituting in the first equation:</p><p><span>$\frac{1}{2} + x - 2x = \frac{2}{5}$</span></p><p><span>$\frac{1}{2} - x = \frac{2}{5}$</span></p><p><span>$x = \frac{1}{2} - \frac{2}{5} = \frac{5-4}{10} = \frac{1}{10} = 0.10$</span></p><p>∴ Answer is (a) 0.10</p>
Correct Answer: a