The Houston Rockets and San Antonio Spurs will play a "best two out of three" series. Assume that Houston has a probability of 1/3 of winning any game.
a. Draw a probability tree showing possible outcomes of the series. Label the branches with appropriate probabilities.
b. What is the probability that Houston wins in two straight games? that San Antonio wins in two straight games?
c. what is the probability that the series goes to three games?
d. what is the probability that Houston wins the series after losing the first game?
e. What is the probability that San Antonio wins the series?
prob(H win) = 1/3 , prob(A win) = 2/3
prob(H loss) = 2/3 , prob(A loss) = 1/3
a) not good on trees in this forum, hard to show
b) Prob (HH) = (1/3)(1/3) = 1/9
Prob(AA) = (2/3)(2/3) = 4/9
c) to go 3 games, could be
AHA or HAH
prob of that = (2/3)(1/3)(2/3) + (1/3)(2/3)(1/3) = 4/27+2/27
= 6/27
= 2/9
d) has to be AHH , showing the winning results
prob(AHH) = (2/3)(1/3)(1/3) = 2/27
e) three ways for Antonio to win the series
1. AA -----> (2/3)(2/3) = 4/9
2. AHA ---->(2/3)(1/3)(2/3) = 4/27
3. HAA ----> (1/2)(2/3)(2/3) = 4/27
prob(A wins series) = 4/9 + 2(4/27) = 20/27
To answer these questions, we need to create a probability tree to visualize the possible outcomes of the series. In a best two out of three series, there are three games and each team can either win or lose each game.
a. Here is the probability tree for the series:
/ Houston (1/3)
Houston -+
\ San Antonio (2/3)
/ Houston (2/3)
San Antonio-+
\ San Antonio (1/3)
/ Houston (2/3)
Houston -+
\ San Antonio (1/3)
This tree shows the three possible outcomes for each game. Each branch is labeled with the probability of that outcome occurring.
b. To determine the probability of Houston winning in two straight games, we follow the branches for Houston, which occur with a probability of 1/3 and 2/3 respectively:
P(Houston winning in two straight games) = 1/3 * 2/3 = 2/9
To determine the probability of San Antonio winning in two straight games, we follow the branches for San Antonio, which occur with a probability of 2/3 and 1/3 respectively:
P(San Antonio winning in two straight games) = 2/3 * 1/3 = 2/9
c. The probability that the series goes to three games is the probability of each team winning one game. This can be calculated by multiplying the probabilities of each team winning a game and then doubling it to account for the two possible orders of wins:
P(series goes to three games) = 2 * (1/3 * 2/3) = 4/9
d. The probability that Houston wins the series after losing the first game is the probability of Houston winning the next two games. This can be calculated by following the branch for Houston losing the first game, then for Houston winning the next two games:
P(Houston wins the series after losing the first game) = (2/3) * (2/3 * 1/3) = 4/27
e. To find the probability that San Antonio wins the series, we need to consider all the possible ways for that to happen. San Antonio can either win in two straight games or lose the first game and then win the next two games. We add these probabilities together:
P(San Antonio wins the series) = P(San Antonio winning in two straight games) + P(San Antonio winning after losing the first game)
= 2/9 + 4/27
= 18/81 + 12/81
= 30/81
= 10/27
Therefore, the probability that San Antonio wins the series is 10/27.