Question

based on past results, a batter knows that the opposing pitcher throws a fastball 75% of the time and a curveball 25% of the time. suppose the batter sees 8 pitches during a particular at - bat. determine each probability. round your answers to the nearest tenth of a percent if necessary. sample problem p(4 fastballs and 4 curveballs) =₈c₄(3/4)⁴(1/4)⁴≈70(0.0012359)≈0.087 8.7% p(all fastballs) > enter the answer in the space provided. use numbers instead of words.

Explanation:

Step1: Identify probability of single - event

The probability of a fast - ball $p = 0.75=\frac{3}{4}$ and the probability of a non - fast - ball (curveball) $q=0.25 = \frac{1}{4}$. We want to find the probability of 8 fast - balls in 8 pitches.

Step2: Use binomial probability formula

The binomial probability formula is $P(X = k)=_{n}C_{k}p^{k}q^{n - k}$, where $n$ is the number of trials, $k$ is the number of successful trials, $p$ is the probability of success on a single trial, and $q$ is the probability of failure on a single trial. Here, $n = 8$, $k = 8$, $p=\frac{3}{4}$, and $q=\frac{1}{4}$. Since $_{n}C_{k}=\frac{n!}{k!(n - k)!}$, when $n = 8$ and $k = 8$, $_{8}C_{8}=\frac{8!}{8!(8 - 8)!}=1$. Then $P(X = 8)=_{8}C_{8}(\frac{3}{4})^{8}(\frac{1}{4})^{0}$.

Step3: Calculate the result

$(\frac{3}{4})^{8}=\frac{3^{8}}{4^{8}}=\frac{6561}{65536}\approx0.1$.

Answer:

$10.0\%$