Median of five random numbers under 0.4

Question

Draw 5 independent numbers uniformly from [0,1]. The sample median is the 3rd smallest (the middle one). What is the probability that this median is below 0.4?

Accepted Answer

The median (3rd of 5) falls below 0.4 exactly when at least 3 of the five numbers land below 0.4. Each lands below 0.4 with probability 0.4, so the count is Binomial(5, 0.4). We need 3, 4, or 5: P = \binom{5}{3}(0.4)^3(0.6)^2 + \binom{5}{4}(0.4)^4(0.6) + \binom{5}{5}(0.4)^5. Over the denominator 5^5 = 3125, the terms are \frac{720}{3125}, \frac{240}{3125}, and \frac{32}{3125}, so P = \frac{720 + 240 + 32}{3125} = \frac{992}{3125} \approx 0.317. Takeaway. With five draws, the median dips below t iff at least three do, so evaluate the upper tail of Binomial(5, t).

Median of five random numbers under 0.4

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