Problem 30 · 2021 Math Kangaroo
Stretch
Counting & Probability
work-backward
A certain game is won when one player gets 3 points ahead. Two players A and B are playing the game and at a particular point, A is 1 point ahead. Each player has an equal probability of winning each point. What is the probability that A wins the game?
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Answer: B — \(\tfrac{2}{3}\)
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Hint 1 of 2
Track A's lead as a walk that ends at +3 (A wins) or −3 (A loses), starting at +1.
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Hint 2 of 2
For a fair walk, the chance of reaching one boundary first is proportional to the distance from the other.
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Approach: model as a symmetric walk between absorbing boundaries
- Let A's lead change by ±1 with equal chance; A wins at +3 and loses at −3, starting from +1.
- For a fair random walk, P(reach +3 before −3) = (start − lower)/(upper − lower) = (1−(−3))/(3−(−3)).
- That is 4/6 = 2/3.
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