Problem 35 · AMC 8 Stretch
Core
Counting & Probability
Logic & Word Problems
and-process-multiplylogical-reasoning
Blake thinks his chance of getting into college A is 0.75 and into college B is 0.5. He multiplies to claim the chance of getting into BOTH is 0.375. Explain why his reasoning might not be right.
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Answer: Multiplying assumes independence, which is doubtful here
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Hint 1 of 3
Multiplying \(P(A) \times P(B)\) for 'A and B' only works under one special condition.
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Hint 2 of 3
What has to be true about the two events for multiplying to be valid?
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Hint 3 of 3
Think about whether getting into one college is really unrelated to getting into the other (his grades, scores, and essays affect both).
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Approach: Multiplying probabilities requires independence
- Multiplying two probabilities to get 'both happen' only works when the two events are INDEPENDENT — one happening has no effect on the chance of the other.
- Blake's arithmetic (\(0.75 \times 0.5 = 0.375\)) is fine, but the multiplication itself is only allowed if the two acceptances are independent.
- In real life they probably are NOT: the same grades, test scores, and essays affect both colleges. A strong applicant tends to get into both, a weaker one tends to be rejected by both.
- So the events are linked, and the true chance of 'both' is likely higher than 0.375. His reasoning isn't justified unless the events are independent, which is doubtful here.
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