Beating the other bid by a wide margin

Question

Two independent scores X and Y are each drawn uniformly between 0 and 10, with no draw affecting the other. What is the probability that the first score beats the second by more than 5, that is, X - Y 5?

Accepted Answer

Drop the pair (x, y) uniformly into a 10-by-10 square; all points are equally likely, so probability equals favorable area over 100. We want the region where x - y 5, meaning x y + 5. This is a one-sided condition (not the symmetric band from the meeting problems), so it carves off a single triangular corner rather than a diagonal strip. The boundary line x = y + 5 runs from the point (5, 0) up to (10, 5). Above-and-right of that line, inside the square, is a right triangle with both legs of length 5: 	ext{area} = \frac{1}{2}(5)(5) = 12.5. Divide by the total area 100: P(X - Y 5) = \frac{12.…

Beating the other bid by a wide margin

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