Question: A baseball analyst assigns subjective estimates for the probability that each team will win their division. He believes there is a 0.3 probability that Boston will win the American League East and a 0.25 probability that Tampa Bay will win the American League East. He believes there is a 0.25 probability that Detroit will win the American League Central.
What is the probability that Boston or Tampa Bay will end up winning the Eastern Division?
What is the probability that Boston or Detroit will end up winning their divisions?
What is the probability that at least one of these three teams — Boston, Tampa Bay, or Detroit — will win their division?
Answer: These problems help explain the concept of mutually exclusive events and independent events.
Two events are said to be mutually exclusive if they cannot both occur at the same time. The event the Boston Red Sox wins their division and the event Tampa Bay wins the division are mutually exclusive because both teams are in the same division.
The probability that at least one of a number of mutually exclusive events occurs is simply the sum of the probabilities of each mutually exclusive outcome. The probability that either Boston or Tampa Bay win the Eastern Division is the P(Boston Wins) + P(Tampa Bay Wins) = 0.30 + 0.25, which is 0.55.
A division championship by Detroit is not mutually exclusive with a division victory by Boston because the two teams are in different divisions. Both teams can win their respective division.
The probability that either Boston or Detroit win their division is
P(Boston wins) + P(Detroit Wins) - P (Both Boston & Detroit Win)
This is 0.3+0.25 - (0.30 x 0.25) or 0.475
Note: A Detroit Title and a Boston Title are independent of each other so the probability that both Detroit wins and Boston wins is the product of the two probabilities.
P(Both Detroit and Boston Win) = P(Detroit Wins) x P(Boston Wins)
The probability of either Detroit and Boston winning is lower than the probability of either Boston and Tampa Bay winning. This result is intuitive. Tampa Bay and Boston are both in the same division. Boston and Detroit are in different divisions. If an analyst picked five teams in the same division to win the division he would have picked the winner with certainty. If the analyst picked two teams from one division to win the division and three teams from another division to win then there is a real possibility that none of his teams wins either division.
Now what is the probability that one of the three teams gets a division title?
We found that the probability that either Boston or Tampa Bay win the East is 0.55.
We were given that that the probability of Detroit winning the Central Division was 0.25.
The outcomes from the two divisions are independent.
The probability that one of the three teams wins their division is (0.55+ 0.25 - 0.25 x 0.55) or 0.6625.