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A low median

MediumProbabilityRandom Variables~5m

Three numbers are drawn independently from Uniform[0,1]\text{Uniform}[0,1]. What is the probability that their median is less than 23\tfrac23?

Show hints (2)+
  1. Median of 33 is <t<t iff at least 22 of the 33 draws are <t<t.
  2. That's 3t2(1−t)+t3=3t2−2t33t^2(1-t)+t^3 = 3t^2-2t^3; plug t=2/3t=2/3.

Answer

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0.7407 (± 0.001)

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Asked at: Jane Street, SIG

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