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 \the probability of getting a vowel on exactly 3 of the spins.\ complete compute p(3 successes) for 5 spins of the spinner. done

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:

\(\frac{40}{243}\)