based on past results, a batter knows that the opposing pitcher throws a fastball 75% of the time and a…

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) = _8c_4(\\(\\frac{3}{4}\\))^4(\\(\\frac{1}{4}\\))^4 ≈ 70(0.0012359) ≈ 0.087 ≈ 8.7% p(no fastballs) > enter the answer in the space provided. use numbers instead of words. %

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) = _8c_4(\\(\\frac{3}{4}\\))^4(\\(\\frac{1}{4}\\))^4 ≈ 70(0.0012359) ≈ 0.087 ≈ 8.7% p(no fastballs) > enter the answer in the space provided. use numbers instead of words. %

Answer

Explanation:

Step1: Identify probability formula

We use the binomial - probability formula (P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}), where (n) is the number of trials, (k) is the number of successes, (p) is the probability of success on a single - trial, and (C(n,k)=\frac{n!}{k!(n - k)!}). Here, (n = 8) (number of pitches), the probability of a fast - ball (p = 0.75), and the probability of a curve - ball (1-p = 0.25). We want to find (P(X = 0)) (no fast - balls).

Step2: Calculate the combination (C(n,k))

When (n = 8) and (k = 0), (C(8,0)=\frac{8!}{0!(8 - 0)!}=\frac{8!}{8!}=1).

Step3: Calculate the probability

Substitute into the binomial - probability formula: (P(X = 0)=C(8,0)\times(0.75)^{0}\times(0.25)^{8}). Since ((0.75)^{0}=1), then (P(X = 0)=1\times1\times(0.25)^{8}). ((0.25)^{8}=\left(\frac{1}{4}\right)^{8}=\frac{1}{65536}\approx0.00001526). As a percentage, (P(X = 0)\approx0.001526%\approx0.0%) (rounded to the nearest tenth of a percent).

Answer:

0.0%