what is the probability of getting a vowel (a success) for the spinner shown? 1/3 complete suppose you spin the spinner 5 times. p(3 successes) means he probability of getting a vowel on exactly 3 of the spins.\ complete compute p(3 successes) for 5 spins of the spinner. done

Question

what is the probability of getting a vowel (a success) for the spinner shown? 1/3 complete suppose you spin the spinner 5 times. p(3 successes) means 	he probability of getting a vowel on exactly 3 of the spins.\ complete compute p(3 successes) for 5 spins of the spinner. done

Accepted Answer

# Explanation:
## Step1: Identify 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 values of $n$, $k$, and $p$
We know that $n = 5$ (number of spins), $k = 3$ (number of successes - getting a vowel), and $p=\frac{1}{3}$ (probability of getting a vowel on a single spin), so $1 - p = 1-\frac{1}{3}=\frac{2}{3}$.
## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(5,3)&=\frac{5!}{3!(5 - 3)!}\
&=\frac{5!}{3!2!}\
&=\frac{5\times4\times3!}{3!\times2\times1}\
& = 10
\end{align*}
$$
## Step4: Calculate $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(5,3)\times p^{3}\times(1 - p)^{5 - 3}\
&=10\times(\frac{1}{3})^{3}\times(\frac{2}{3})^{2}\
&=10\times\frac{1}{27}\times\frac{4}{9}\
&=\frac{40}{243}\approx0.165
\end{align*}
$$
# Answer:
$\frac{40}{243}$

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Question

Explanation:

Step1: Identify binomial - probability formula

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

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

Step4: Calculate \(P(X = 3)\)

Answer: