<p>The odds against a certain event are 5:2 and the odds in favor of other event, independent of the former, are 6:5. Then find the probability that at least one of the events will happen.</p>
Step-by-Step Solution
Key Concept: Convert odds to probabilities: odds against a:b means P(event) = b/(a+b), and odds in favor a:b means P(event) = a/(a+b). Then use P(at least one) = 1 - P(both fail) for independent events.
<p><strong>Step 1: Convert odds to probabilities</strong></p><p>Odds against event A are 5:2, so P(A) = 2/(5+2) = 2/7</p><p>Therefore, P(A') = 5/7</p><p><strong>Step 2: Convert odds in favor to probability</strong></p><p>Odds in favor of event B are 6:5, so P(B) = 6/(6+5) = 6/11</p><p>Therefore, P(B') = 5/11</p><p><strong>Step 3: Apply independence for "at least one"</strong></p><p>Since A and B are independent: P(at least one happens) = 1 - P(both fail)</p><p>P(A' ∩ B') = P(A') × P(B') = (5/7) × (5/11) = 25/77</p><p><strong>Step 4: Calculate final answer</strong></p><p>P(at least one) = 1 - 25/77 = (77 - 25)/77 = 52/77</p>
Correct Answer: 52/77