Question Find the probability that a baseball player with a batting average of $0.33$ has exactly 4 hits in their next 7 at-bats.

Studdy Solution

STEP 1

Assumptions
1. The batting average is 0.33, which means the probability of getting a hit is 0.33.
2. The number of at bats is 7.
3. The number of hits we are interested in is exactly 4.
4. Each at bat is an independent event.

STEP 2

We will use the binomial probability formula to calculate the probability of exactly 4 hits in 7 at bats. The binomial probability formula is given by:
$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
where: - $P(X=k)$ is the probability of getting exactly $k$ successes in $n$ trials, - $\binom{n}{k}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials, - $p$ is the probability of success on a single trial, - $(1-p)$ is the probability of failure on a single trial.

STEP 3

Now, we will calculate the binomial coefficient $\binom{n}{k}$ for $n=7$ and $k=4$ .
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

STEP 4

Calculate the factorial of $n$ , $k$ , and $(n-k)$ .
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ $4! = 4 \times 3 \times 2 \times 1$ $3! = 3 \times 2 \times 1$

STEP 5

Now, we plug in the values to find the binomial coefficient.
$\binom{7}{4} = \frac{7!}{4! \times (7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1}$

STEP 6

Simplify the binomial coefficient.
$\binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

STEP 7

Now, we will calculate the probability of exactly 4 hits using the batting average (probability of success) $p=0.33$ .
$P(X=4) = \binom{7}{4} \times 0.33^4 \times (1-0.33)^{7-4}$

STEP 8

Calculate the probability of exactly 4 hits.
$P(X=4) = 35 \times 0.33^4 \times (1-0.33)^3$

STEP 9

Simplify the calculation by computing the powers and the multiplication.
$P(X=4) = 35 \times (0.33)^4 \times (0.67)^3$

STEP 10

Perform the exponentiation and multiplication to find the probability.
$P(X=4) = 35 \times (0.33 \times 0.33 \times 0.33 \times 0.33) \times (0.67 \times 0.67 \times 0.67)$

STEP 11

Complete the calculation to get the final probability.
$P(X=4) = 35 \times 0.01188181 \times 0.300763$

STEP 12

Multiply the values to get the final probability.
$P(X=4) \approx 35 \times 0.01188181 \times 0.300763 \approx 0.1254$
The probability that the baseball player has exactly 4 hits in his next 7 at bats is approximately 0.1254.

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