Question

what is the probability of getting a vowel (a success) for the spinner shown? suppose you spin the spinner 5 times. means \the probability of exactly 3 of the spins.\ p(3 successes) p(4 successes) p(5 successes)

Explanation:

Step1: Identify probability of single - success

The spinner has 3 equal - sized sections. There is 1 vowel (A) out of 3 sections. So the probability of getting a vowel in a single spin, $p=\frac{1}{3}$, and the probability of not getting a vowel, $q = 1 - p=1-\frac{1}{3}=\frac{2}{3}$.

Step2: Use binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times q^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success in a single trial, $q$ is the probability of failure in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 5$.
We want to find the probability of exactly 3 successes.
First, calculate the combination $C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5!}{3!2!}=\frac{5\times4}{2\times1}=10$.
Then, $p=\frac{1}{3}$, $q=\frac{2}{3}$, $n = 5$, $k = 3$.
$P(X = 3)=C(5,3)\times(\frac{1}{3})^{3}\times(\frac{2}{3})^{2}$.
$P(X = 3)=10\times\frac{1}{27}\times\frac{4}{9}=\frac{40}{243}$.

Answer:

$P(3\text{ successes})=\frac{40}{243}$