Two equally matched teams are playing a game of volleyball. Each team has a 1/2 chance of winning each point. The game ends once one of the teams gets to a score of 21. If the score is currently 18-16 for team A, the probability that team A will win the game can be expressed as a/b where a and b are coprime positive integers. What is the value of a+b?
Details and Assumptions:
Clarification: Team A currently has 18 points, expressed via "18-16 for team A".
11 years ago
11 months ago
To find the probability that team A will win the game given that the score is currently 18-16 for team A, we need to consider the possible ways the game can end.
Team A can win in the next two points, in which case the final score will be 20-16, and they will have won the game. This has a probability of (1/2)^2 = 1/4.
Team B can win in the next five points, in which case the final score will be 18-21, and they will have won the game. This also has a probability of (1/2)^5 = 1/32.
Therefore, the probability of team A winning the game, given the current score, is 1/4 + 1/32 = 9/32.
The values of a and b in the fraction a/b that represents the probability are a = 9 and b = 32.
To find the value of a+b, we just add the two numbers together: a+b = 9+32 = 41.
Therefore, the value of a+b is 41.