Probability I · Probability spaces, axioms, and rigorous counting
Choose a three-element subset uniformly from {1,…,9}, and let M be its largest element
Problem
Choose a three-element subset uniformly from \(\{1,\ldots,9\}\), and let \(M\) be its largest element. Count the subsets with \(M=k\) for every possible \(k\), and find \(\Pr(M\le6)\) and \(\Pr(M=9)\). Use the largest-element argument to prove \[ \sum_{j=r}^{N}\binom jr=\binom{N+1}{r+1} \] for integers \(N\ge r\ge0\).
Hint
Once the largest element is fixed, it is no longer a choice.
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