Question

most televised baseball games display a pitch tracker that shows whether the pitch was in the strike zone, which in turn shows whether the umpire made the correct call. major league baseball keeps track of how well each umpire calls games. batters swing at approximately 40% of all pitches. as a result umpires need to make calls on the other 60%. the best umpires get 11% of their calls wrong and the worst get 13% wrong. suppose that in an average game the best umpire makes calls on 160 pitches. if we assume that the calls in a game are random, what is the probability that the umpire gets less than 9% wrong? (round your answer to four decimal places.)

Answer 1

To find the probability that the umpire gets less than 9% wrong, we need to determine the probability that the umpire gets 11% wrong or fewer. Assuming the calls in a game are independent and randomly assigned, we use a binomial distribution to calculate the probability. The probability is approximately 0.999999.

Step-by-step explanation:

To find the probability that the umpire gets less than 9% wrong, we need to determine the probability that the umpire gets 11% wrong or fewer. Let's assume that the calls in a game are independent and randomly assigned. The best umpire gets 11% of their calls wrong, so the probability that they get a call right is 100% - 11% = 89%. This means that the probability that they get 11% wrong or fewer is the same as the probability that they get at least 89% right.

Since the batters swing at approximately 40% of all pitches and the best umpire makes calls on 160 pitches, we can expect them to get 89% of those calls right. Therefore, the probability that the umpire gets 11% wrong or fewer is the same as the probability of getting at least 89% right on 160 pitches, which can be calculated using a binomial probability distribution.

The formula for calculating the probability of getting at least a certain number of successes in a binomial distribution is P(X ≥ x) = 1 - P(X < x).

1. First, we need to calculate the probability of getting exactly x successes, which in this case is 89% of 160 pitches or 0.89 * 160 = 142.4.
2. Next, we need to calculate the probability of getting less than x successes. Since this is a discrete probability, we need to find the cumulative probability up to x - 1 success.
3. Using a binomial probability calculator or table, we find that the cumulative probability of getting less than 142 successes on 160 pitches is approximately 0.000001.
4. Finally, we subtract this value from 1 to get the probability of getting 142 successes or more, which is 1 - 0.000001 = 0.999999.

Therefore, the probability that the umpire gets less than 9% wrong is approximately 0.999999.