Question

1. an archer is able to hit the bulls - eye 46% of the time. if she shoots 9 arrows, what is the probability that she gets exactly 4 bulls - eyes? assume each shot is independent of the others. a) 0.0448 b) 0.259 c) 0.203 d) 0.00206

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Determine the values of $n$, $k$, and $p$

Here, $n = 9$ (number of arrows shot), $k = 4$ (number of bull's - eyes), and $p=0.46$ (probability of hitting the bull's - eye in a single shot), and $1 - p = 1-0.46 = 0.54$.

Step3: Calculate the combination $C(n,k)$

$C(9,4)=\frac{9!}{4!(9 - 4)!}=\frac{9!}{4!5!}=\frac{9\times8\times7\times6}{4\times3\times2\times1}=126$.

Step4: Calculate the probability

$P(X = 4)=C(9,4)\times(0.46)^{4}\times(0.54)^{9 - 4}=126\times(0.46)^{4}\times(0.54)^{5}$.
$(0.46)^{4}=0.46\times0.46\times0.46\times0.46\approx0.0447$, $(0.54)^{5}=0.54\times0.54\times0.54\times0.54\times0.54\approx0.0459$.
$P(X = 4)=126\times0.0447\times0.0459\approx0.259$.

Answer:

B. 0.259