Question

A baseball player has a lifetime batting average of 0.233. If, in a season, this player has 420 "at bats", what is the probability he gets 84 or more hits

Answer 1

To calculate the probability that the baseball player gets 84 or more hits in a season, we use the binomial probability formula. The resulting probability will give us the answer to the question.

Step-by-step explanation:

To calculate the probability that the baseball player gets 84 or more hits in a season, we need to use the binomial probability formula. The formula is P(X ≥ k) = 1 - P(X < k), where X is the number of hits and k is 84. The probability of getting exactly k hits can be calculated using the binomial probability formula P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where n is the number of at bats, k is the number of hits, and p is the batting average. Let's calculate the probability:

1. n = 420 (number of at bats)
2. p = 0.233 (batting average)
3. k = 84 (number of hits)
4. P(X = k) = C(420, 84) * 0.233^84 * (1 - 0.233)^(420 - 84)
5. P(X < k) = sum of P(X = 0), P(X = 1), ..., P(X = k-1)
6. P(X ≥ 84) = 1 - P(X < 84)

Now let's calculate the probability using the given values:

1. C(420, 84) = 420! / (84! * (420 - 84)!)
2. P(X = k) = C(420, 84) * 0.233^84 * (1 - 0.233)^(420 - 84)
3. P(X < k) = sum of P(X = 0), P(X = 1), ..., P(X = 83)
4. P(X ≥ 84) = 1 - P(X < 84)

The resulting probability will give us the answer to the question.