2×2 Matrix With Entries 0 or 1 — Variance of the Determinant
A 2×2 matrix has entries from {0,1}. Let X be its determinant. Find the variance of X by listing all 16 matrices and their determinant probabilities.
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16-Square Board: Probability Two Random Squares Share No Side
A 4×4 board has 16 squares. Two are chosen at random. Find the probability they share no side, by counting adjacent pairs directly.
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A and B Throw Dice Alternately, A Wins on Sum 5 Before B’s Sum 8
A and B alternately throw a pair of dice; A wins by throwing sum 5 before B throws sum 8. Full solution using independent events and an infinite GP.
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Committee of 4 From 8 Boys, 4 Girls — P(Exactly 2 Girls | At Least 1 Girl)
A committee of 4 is chosen from 8 boys and 4 girls. Given at least one girl is selected, find the probability of exactly 2 girls, using the conditional probability formula.