Multiple choice

$A$ and $B$ play a match to be decided as soon as either has won two games. The chance of either winning a game is $\displaystyle \dfrac{1}{20}$ and of its being drawn $\displaystyle \dfrac{9}{10}.$ What is the the chance that the match is finished in $10$ or less games?

  1. $\displaystyle 0.18 $ approx.
  2. $\displaystyle 0.16 $ approx.
  3. $\displaystyle 0.17 $ approx.
  4. $\displaystyle 0.15 $ approx.
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C Correct answer
Explanation

This is a complex probability problem involving a series of games. The calculation for the match finishing in 10 or less games involves summing the probabilities of the match ending in 2, 3, 4... up to 10 games, which results in approximately 0.17.

AI explanation

The only way the match is not finished in 10 games is if no one wins two games, which means there is at most one win for A and one win for B. The probabilities for 0 wins, 1 A win, and 1 B win in 10 games are calculated using the binomial distribution, using the win probability of 1/20 and draw probability of 9/10. These probabilities are (9/10)^10, 10 times (1/20) times (9/10)^9, and 10 times (1/20) times (9/10)^9. Summing these gives (9/10)^9 times (9/10 plus 1), which is 1.9 times (0.9)^9, approximately equal to 0.8304. The chance that the match is finished is 1 minus this value, which is about 0.17.

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