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# Counting and Probability

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Bob and Meena play a two-person game which is won by the first person to accumulate 10 points. At each turn Bob gains a point with probability of 1/3. If he doesn't get a point, then Meena gets a point. Meena is now ahead 9 to 8. What is the probability that Meena will win? Express your answer as a common fraction.

Jul 27, 2022

#1
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The probability that Meena wins is 21/25

In order for Meena to win, she needs to win the next turn or the following one, otherwise, she loses. The probability for that is equal to subtracting from 1 the probability of the complementary event: bob wins in the next 2 turns. Since each turn is independent of the other, we can obtain the probability of Bob winning the next 2 turns by taking the square of the probability of him winning on one turn, hence it is

Thus, the probability for Meena to win is 1-4/25 = 21/25.

Jul 27, 2022
#2
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The only way Meena loses is if Bob wins the next 2 points. The probability of Bob winning the next 2 points is $$({1 \over 3})^2 = {1 \over 9}$$.

So, the probability Meena wins is $$1 - {1 \over 9} = \color{brown}\boxed{8 \over 9}$$

Jul 27, 2022