Question

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. 0.133
c. 0.00184
d. 0.0635

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 $C(n,k)$

$C(9,3)=\frac{9!}{3!(9 - 3)!}=\frac{9!}{3!6!}=\frac{9\times8\times7}{3\times2\times1}=84$.

Step4: Calculate the probability $P(X = 3)$

$P(X = 3)=C(9,3)\times(0.51)^{3}\times(0.49)^{6}$
$P(X = 3)=84\times(0.51)^{3}\times(0.49)^{6}$
$(0.51)^{3}=0.51\times0.51\times0.51 = 0.132651$
$(0.49)^{6}=0.49\times0.49\times0.49\times0.49\times0.49\times0.49\approx0.0168$
$P(X = 3)=84\times0.132651\times0.0168\approx0.184$

Answer:

There seems to be an error in the provided options as the calculated value is approximately $0.184$. If we assume there was a calculation error in the above steps and re - calculate more precisely:

$C(9,3)=\frac{9!}{3!(9 - 3)!}=\frac{9\times8\times7}{3\times2\times1}=84$

$p = 0.51$, $1 - p=0.49$

$P(X = 3)=84\times(0.51)^{3}\times(0.49)^{6}$

$(0.51)^{3}=0.132651$

$(0.49)^{6}=0.0168073$

$P(X = 3)=84\times0.132651\times0.0168073\approx0.184$

If we consider the closest option to our calculated value, none of the options match exactly. But if we assume some rounding differences in the problem - setup calculations, the closest value to our result among the given options is A. 0.154 (although it is not an exact match). So, if we have to choose from the given options, the answer is A. 0.154.