Conditional ordering of three iid uniforms

Question

Let $X_1,\dots,X_{20}$ be independent $\text{Unif}(0,1)$ random variables. Given that $X_{10}<X_{20}$ , what is the probability that $X_1>X_{10}$ ?

Short answer

$2/3$ . Both conditions together mean that, among the three variables with indices $1$ , $10$ and $20$ , the tenth is the minimum; by exchangeability that has probability $\tfrac13$ . Dividing by $P(X_{10}<X_{20})=\tfrac12$ gives $\tfrac23$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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