All three random numbers below one half

Question

Draw 3 independent numbers uniformly from [0,1] and let M be the largest. What is the probability that M is below 0.5?

Accepted Answer

The largest is below 0.5 exactly when all three numbers are below 0.5. Each is below 0.5 with probability 0.5, and they are independent, so P(M < 0.5) = 0.5^3 = \frac{1}{8} = 0.125. So it is only a one-in-eight chance that all three draws avoid the upper half of the interval. Takeaway. The maximum being small forces every draw to be small: P(M < t) = t^n.

All three random numbers below one half

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