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counting and probability senior
Problem
If six people decide to come to a basketball game, but three of them are only 2/5 sure that they will stay for the entire time (the other three are sure they'll stay the whole time), what is the probability that at the end, at least 5 people stayed the entire time?
Solution
There are two cases: 5 people and 6 people stayed.
Case 1: 5 people stayed the whole time. The probability that exactly 2 of those that are unsure stayed the entire time is .
Case 2: 6 people stayed the whole time. The probability that all three unsure people stayed is .
The sum of these probabilities is .
Case 1: 5 people stayed the whole time. The probability that exactly 2 of those that are unsure stayed the entire time is .
Case 2: 6 people stayed the whole time. The probability that all three unsure people stayed is .
The sum of these probabilities is .
Final answer
\frac{44}{125}