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# Probability and Statistics

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A high school baseball player has a 0.181 batting average. In one game, he gets 8 at bats. What is the probability he will get at least 5 hits in the game?

Mar 11, 2023

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To find the probability that the baseball player will get at least 5 hits in the game, we can use the binomial distribution, since the player either gets a hit or does not get a hit on each at bat, and the at bats are independent.

Let X be the number of hits the player gets in the game. Then X follows a binomial distribution with parameters n = 8 (the number of at bats) and p = 0.181 (the probability of getting a hit on each at bat).

The probability of getting at least 5 hits is equal to the sum of the probabilities of getting 5, 6, 7, or 8 hits:

P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

Using the formula for the probability mass function of the binomial distribution, we can calculate each of these probabilities:

P(X = k) = (n choose k) p^k (1-p)^(n-k)

where (n choose k) is the binomial coefficient.

Therefore, we have:

P(X = 5) = (8 choose 5) (0.181)^5 (1-0.181)^(8-5) ≈ 0.222

P(X = 6) = (8 choose 6) (0.181)^6 (1-0.181)^(8-6) ≈ 0.079

P(X = 7) = (8 choose 7) (0.181)^7 (1-0.181)^(8-7) ≈ 0.013

P(X = 8) = (8 choose 8) (0.181)^8 (1-0.181)^(8-8) ≈ 0.001

Therefore, the probability of getting at least 5 hits in the game is:

P(X ≥ 5) = 0.222 + 0.079 + 0.013 + 0.001 = 0.315

Therefore, the probability that the baseball player will get at least 5 hits in the game is approximately 0.315 or 31.5% (rounded to one decimal place).

Mar 11, 2023