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Where the maximum lands

EasyRandom VariablesStatistics~3m

X1,…,X5X_1,\dots,X_5 are independent uniform [0,1][0,1]. What is the probability that their maximum lies between 0.50.5 and 0.80.8, i.e. P(0.5≤max⁡iXi≤0.8)P\big(0.5 \le \max_i X_i \le 0.8\big)? (to 4 decimals)

Show hints (2)+
  1. CDF of the max of n iid uniforms is F(x) = xⁿ.
  2. P(a ≤ max ≤ b) = F(b) − F(a) = 0.8⁵ − 0.5⁵.

Answer

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0.2964 (± 0.0005)

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Asked at: Timed Mental-Math & Sequences, Options Market-Making

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