Joe's batting average is .323. (That is, he averages 0.323 hits per at bat.) What is the probability that he will get three hits in three at-bats? Express your answer as a decimal to the nearest hundredth.
To find the probability that Joe will get three hits in three at-bats, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X = k) is the probability of getting exactly k hits.
n is the number of trials (at-bats in this case), which is 3.
k is the number of successful outcomes (hits in this case), which is also 3.
p is the probability of success on a single trial (Joe's batting average), which is 0.323.
(n choose k) represents the binomial coefficient, which is calculated as n! / (k!(n - k)!).
Let's calculate it step by step:
Calculate (n choose k): (3 choose 3) = 3! / (3!(3 - 3)!) = 1
Calculate p^k: 0.323^3 ≈ 0.033857
Calculate (1 - p)^(n - k): (1 - 0.323)^(3 - 3) = 0.677911
Now, plug these values into the formula:
P(X = 3) = 1 * 0.033857 * 0.677911 ≈ 0.02299
So, the probability that Joe will get three hits in three at-bats is approximately 0.023, rounded to the nearest hundredth.