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.
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 binomial probability formula is given by:
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
- x is the number of successes
- p is the probability of success on a single trial
- q is the probability of failure on a single trial (1 - p)
In this case, the baseball player's batting average is given as 0.395, which means the probability of getting a hit (success) on a single at-bat is 0.395. Therefore, p = 0.395.
The baseball player has 7 at-bats (trials) and we want to find the probability of exactly 3 hits, so x = 3 and n = 7.
Substituting these values into the binomial probability formula, we get:
P(3) = (7C3)(0.395^3)(0.605^(7-3))
To evaluate this expression, let's break it down step by step:
1. Calculate the number of combinations:
(7C3) = 7! / (3! * (7-3)!) = 35
2. Substitute the values into the formula:
P(3) = 35 * (0.395^3) * (0.605^4)
3. Calculate the probabilities:
P(3) = 35 * 0.058509875 * 0.084774125
4. Multiply the values:
P(3) = 0.17585
Finally, round the answer to 4 decimal places as requested:
The probability that the baseball player has exactly 3 hits in his next 7 at-bats is approximately 0.1759.