A baseball player has a batting average of 0.395. What is the probability that he has exactly 3 hits in his next 7 at bats? Round your answer to 4 decimals.
The probability is
To find the probability that the baseball player has exactly 3 hits in his next 7 at-bats, we can use the binomial probability formula. The formula is:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials
- p is the probability of a single success
- q is the probability of a single failure (1 - p)
- nCx is the binomial coefficient, which represents the number of ways to choose x successes from n trials
In this case, the baseball player has 7 at-bats, so n = 7. The probability of getting a hit in a single at-bat is 0.395, so p = 0.395. The probability of not getting a hit in a single at-bat is 1 - p = 1 - 0.395 = 0.605, so q = 0.605. We want to find the probability of getting exactly 3 hits, so x = 3.
Now, we can plug these values into the formula:
P(3) = (7C3) * (0.395^3) * (0.605^(7-3))
To calculate (7C3), we use the formula:
nCx = n! / (x! * (n-x)!), where ! denotes the factorial.
(7C3) = 7! / (3! * (7-3)!) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
Now, substitute all the values into the formula:
P(3) = 35 * (0.395^3) * (0.605^4)
Calculating this expression, we get:
P(3) = 0.0271
Therefore, the probability that the baseball player has exactly 3 hits in his next 7 at-bats is approximately 0.0271, rounded to 4 decimals.