Probability
Event Combinations
Grade 12
Question:
<p>For three events A, B and C, denote:<br/>P(Exactly one of A or B or C occurs) = \(P_1\)<br/>P(Exactly one of B or C occurs) = \(P_2\)<br/>P(Exactly one of C or A occurs) = \(P_3\)<br/>P(All the three events occur simultaneously) = \(\frac{1}{16}\)<br/>If \(P_1 = P_2 = P_3 = \frac{1}{4}\), then the probability that at least one of the events occurs, is [JEE Main 2017, 4M]</p>
<p>(a) \(\frac{3}{16}\)</p>
<p>(b) \(\frac{7}{32}\)</p>
<p>(c) \(\frac{7}{16}\)</p>
<p>(d) \(\frac{7}{64}\)</p>
Step-by-Step Solution
Key Concept: Set up equations from the given probability conditions about exactly one event occurring, then solve for individual event probabilities.
<p>Let $P(A) = a$, $P(B) = b$, $P(C) = c$, and assume independence. Then P(Exactly one of A or B or C) means exactly one occurs and exactly two don't. The constraint equations along with $P(A \cap B \cap C) = \frac{1}{16}$ lead to determining individual probabilities. At least one occurs = $1 - P(\text{none})$.</p>
Correct Answer: C