Median of three random numbers below 0.25

Question

Draw 3 independent numbers uniformly from [0,1]. The sample median is the 2nd smallest (the middle one). What is the probability that this median is below 0.25?

Accepted Answer

The median (2nd of 3) falls below 0.25 exactly when at least 2 of the three numbers land below 0.25. Each lands below 0.25 with probability 0.25, so the count below 0.25 is Binomial(3, 0.25). We need 2 or 3: P = \binom{3}{2}\left(	frac14ight)^2\left(	frac34ight) + \binom{3}{3}\left(	frac14ight)^3 = \frac{9}{64} + \frac{1}{64} = \frac{10}{64} = \frac{5}{32} \approx 0.156. Takeaway. "Median below t" mirrors "median above t": it needs at least half the sample below t, again a binomial tail.

Median of three random numbers below 0.25

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