Problem Solving

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.Find the probability that Adam wins the game by scoring not more than four more points.
OPTIONS
A.8/27
B.40/81
C.16/81
D.88/243
E.None of these
OA Option B

firdaus117 wrote: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.Find the probability that Adam wins the game by scoring not more than four more points.
OPTIONS
A.8/27
B.40/81
C.16/81
D.88/243
E.None of these
OA Option B

Adam can win the game in the following different possibilities, because just 1 or 2 win won't make him a possible winner here:

Possibility # 1

Adam wins 3 games on a trot, and its probability = (2/3) ^3 = 8/27.

Possibility # 2

There is only one way how both of the players may reach on the 20 mark and Adam wins, finally. In terms of Adam, the sequence MUST be one of the kinds (WLW) WW. The bracket contents are worth taking care about. WLW can be had in 3! /2! = 3 number of ways, and its probability = (2/3) ^2 × 1/3 × 3 × (2/3) ^2 = 16/81.

Hence, the total probability = 8/27 + 16/81 = [spoiler]40/81[/spoiler].

[spoiler]B[/spoiler]