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. Here is a probability tree showing possible outcomes of the series:
/------------------------ Houston (1/3)
/
--------|--------- San Antonio (2/3)
|
\------------------------ San Antonio (2/3)
|
\------------------------ Houston (1/3)
b. The probability that Houston wins in two straight games is (1/3) * (1/3) = 1/9.
The probability that San Antonio wins in two straight games is (2/3) * (2/3) = 4/9.
c. The probability that the series goes to three games is 1 - (1/9 + 4/9) = 4/9.
d. To find the probability that Houston wins the series after losing the first game, we need to consider two scenarios:
Scenario 1: Houston wins the second and third game.
Scenario 2: San Antonio wins the second game and Houston wins the third game.
Probability of Scenario 1 = (2/3) * (1/3) = 2/9.
Probability of Scenario 2 = (1/3) * (2/3) * (1/3) = 2/27.
Therefore, the probability that Houston wins the series after losing the first game is 2/9 + 2/27 = 8/27.
e. The probability that San Antonio wins the series is 1 - (1/9 + 8/27) = 14/27.
a. To draw a probability tree showing the possible outcomes of the series, consider the following:
/ Houston (1/3)
/
Start - +
\
\ San Antonio (2/3)
At the start of the series, there are two branches: one for Houston winning (with a probability of 1/3) and one for San Antonio winning (with a probability of 2/3).
b. The probability that Houston wins in two straight games is given by the product of the probability of winning the first game (1/3) and the probability of winning the second game (1/3):
P(Houston wins in two straight games) = (1/3) * (1/3) = 1/9
Similarly, the probability that San Antonio wins in two straight games is given by the product of the probability of San Antonio winning the first game (2/3) and the probability of San Antonio winning the second game (2/3):
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 the complement of the probabilities of Houston or San Antonio winning in two straight games. So we subtract the probabilities of both teams winning in two straight games from 1:
P(series goes to three games) = 1 - P(Houston wins in two straight games) - P(San Antonio wins in two straight games)
P(series goes to three games) = 1 - (1/9) - (4/9) = 4/9
d. The probability that Houston wins the series after losing the first game can be calculated using conditional probability. This is the probability of Houston winning the second and third games given that they lost the first game.
P(Houston wins the series after losing the first game) = P(Houston wins second game) * P(Houston wins third game | Houston lost first game)
Given that Houston lost the first game, their probability of winning the second game remains unchanged at 1/3. However, they need to win both the second and third games to win the series, so the conditional probability is (1/3) * (1/3) = 1/9.
e. The probability that San Antonio wins the series is the sum of the probabilities of San Antonio winning in two straight games and the probability of the series going to three games with the final outcome being a San Antonio win:
P(San Antonio wins the series) = P(San Antonio wins in two straight games) + P(series goes to three games) * P(San Antonio wins in three games)
P(San Antonio wins the series) = (4/9) + (4/9) * (2/3) = 4/9 + 8/27 = 20/27