At least one large draw among five

Question

Five independent numbers are each drawn uniformly between 0 and 1, with no draw affecting the others. What is the probability that at least one of the five draws exceeds 0.8?

Accepted Answer

"At least one" problems are almost always easier through the back door: find the chance that the event fails everywhere, then subtract from 1. The opposite of "at least one draw above 0.8" is "every draw is at or below 0.8." For a single uniform draw on 0 to 1, the chance of staying at or below 0.8 is 0.8. With five independent draws, all staying below 0.8 has probability P(	ext{all} \le 0.8) = (0.8)^5 = 0.32768. So the chance that at least one pokes above 0.8 is the complement: P(	ext{at least one} 0.8) = 1 - (0.8)^5 = 1 - 0.32768 = 0.67232 \approx 0.672. Even though a single draw beats 0.…

At least one large draw among five

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