Probability
Basic Probability
Grade 12
Question:
<p>In a certain population, 10% of the people are rich, 5% are famous, and 3% are rich and famous. Find the probability that a person picked at random from the population is either famous or rich but not both.</p>
Step-by-Step Solution
Key Concept: Use the principle of inclusion-exclusion to find people who are either rich or famous, then subtract those who are both to get the exclusive-or probability.
<p><strong>Step 1:</strong> Define events and given probabilities.</p><p>Let R = event of being rich, F = event of being famous.</p><p>P(R) = 0.10, P(F) = 0.05, P(R∩F) = 0.03</p><p><strong>Step 2:</strong> Find probability of either rich or famous (inclusive-or).</p><p>P(R∪F) = P(R) + P(F) - P(R∩F) = 0.10 + 0.05 - 0.03 = 0.12</p><p><strong>Step 3:</strong> Find probability of either rich or famous but NOT both (exclusive-or).</p><p>We need: (Rich and not famous) OR (Famous and not rich)</p><p>P(R⊕F) = P(R∪F) - P(R∩F) = 0.12 - 0.03 = 0.09</p><p><strong>Alternatively:</strong> P(R⊕F) = [P(R) - P(R∩F)] + [P(F) - P(R∩F)] = (0.10 - 0.03) + (0.05 - 0.03) = 0.07 + 0.02 = 0.09</p><p>∴ Answer: <strong>0.09</strong></p>
Correct Answer: 0.09