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?
a. To draw a probability tree showing the possible outcomes of the series, we start by representing each game as a branch. Since Houston has a probability of 1/3 of winning any game, we can create the following probability tree:
S
/ \
H(1/3) S(2/3)
/ \ / \
H(1/3) S(2/3) H(1/3)
| | |
Win(1/9) Win(2/9) Win(4/9)
We label the branches with the corresponding probabilities. The tree starts with the first game, with H representing a Houston win and S representing a San Antonio win. Each subsequent level of the tree represents the outcome of the next game. The final level represents the outcome of the series.
b. To find the probability that Houston wins in two straight games, we look at the branch where Houston wins both games (H - H). In this case, the probability is given by multiplying the individual probabilities of Houston winning both games:
P(Houston wins in two straight games) = (1/3) * (1/3) = 1/9
Similarly, to find the probability that San Antonio wins in two straight games, we look at the branch where San Antonio wins both games (S - S). In this case, the probability is given by multiplying the individual probabilities of San Antonio winning both games:
P(San Antonio wins in two straight games) = (2/3) * (2/3) = 4/9
c. The probability that the series goes to three games is given by the branch where each team wins one game (H - S - H). To calculate this probability, we multiply the individual probabilities of each outcome:
P(Series goes to three games) = (1/3) * (2/3) * (1/3) = 2/27
d. To find the probability that Houston wins the series after losing the first game, we look at the branch where Houston loses the first game but wins the next two games (S - H - H). In this case, the probability is given by multiplying the individual probabilities of each outcome:
P(Houston wins the series after losing the first game) = (2/3) * (1/3) * (1/3) = 2/27
e. The probability that San Antonio wins the series is the sum of the probabilities of all possible outcomes where San Antonio wins:
P(San Antonio wins the series) = P(S - S) + P(S - H - S) = (4/9) + (2/27) = 14/27