a tennis player makes a successful first serve 51% of the time. if she serves 9 times, what is the probability that she gets exactly 3 successful first serves in? assume that each serve is independent of the others.
a. 0.154
b. 0.00184
c. 0.133
d. 0.0635

Question

Accepted Answer

# 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, $n = 9$ (the number of serves), $k = 3$ (the number of successful first serves), and $p=0.51$ (the probability of a successful first serve). Then $1 - p=1 - 0.51 = 0.49$

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(9,3)&=\frac{9!}{3!(9 - 3)!}\
&=\frac{9!}{3!6!}\
&=\frac{9\times8\times7}{3\times2\times1}\
&=84
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.51)^{3}=0.51\times0.51\times0.51 = 0.132651$
$(1 - p)^{n - k}=(0.49)^{6}$
$$
\begin{align*}
(0.49)^{6}&=(0.49)^{2}\times(0.49)^{2}\times(0.49)^{2}\
&=(0.2401)\times(0.2401)\times(0.2401)\
&=0.0138412876
\end{align*}
$$

## Step5: Calculate the probability $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(9,3)\times p^{3}\times(1 - p)^{6}\
&=84\times0.132651\times0.0138412876\
&\approx0.154
\end{align*}
$$

# Answer:
A. $0.154$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial probability formula

Step2: Determine the values of \(n\), \(k\), and \(p\)

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

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step5: Calculate the probability \(P(X = 3)\)

Answer: