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Adam wins the game

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by sanju09 » Mon Mar 12, 2012 2:42 am
Adam and Smith are playing a game of badminton. Adam is currently on 18 points and Smith is on 19 points. In badminton, a player wins if he reaches 21 points and the other player has at least 2 points less than him. If both players reach 20 points, then the game continues till there is difference of 2 points between the two players. The probability of Adam winning a point against Smith is 2/3. What is the probability that Adam wins the game by scoring not more than four more points?
A.8/27
B.40/81
C.16/81
D.88/243
E.108/243
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by Neo Anderson » Mon Mar 12, 2012 9:58 am
IMO B 40/81
Adam wins the game by scoring not more than four more points
can be two ways:-
1. Adam wins three points in line => (2/3)*(2/3)*(2/3) = 8/27
2. Adam loses one point in ANY OF THE next three serves {3*(1/3)*(2/3)*(2/3)}, and hence has to win two more points (2/3)*(2/3)(thus making total four points won over smith) => 3*(1/3)*(2/3)*(2/3)*(2/3)*(2/3) = 16/81
total ways 8/27 + 16/81 = 40/81.
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