Solution: We are selecting 5 distinct integers independently and uniformly at random from $ \{0, 1, 2, \ldots, 50\} $, and let $ z $ be the sum of the two largest. We want $ \mathbb{P}(z \geq 100) $.

Fri, 01 May 2026 03:56:36 +0000

Unlocking the Probability: Calculating $ \\mathbb{P}(z \\geq 100) $ for the Sum of Two Largest Integers Chosen from $ \\{0, 1, \\ldots, 50\\} $ Selecting 5 distinct integers independently and uniformly at random from the set $ \\{0, 1, 2, \\ldots, 50\\} $ is a classic problem in discrete probability. A key challenge in such sampling draws is analyzing the sum $ z $ of the two largest values—commonly used in chance operations and risk modeling. This article explores an elegant yet nuanced solution to the question:
What is $ \\mathbb{P}(z \\geq 100) $, where $ z $ is the sum of the two largest of 5 uniformly chosen distinct integers from $ 0 $ to $ 50 $? --- ### The Combinatorial Setting