Question

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

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 $p$

We know that $n = 5$ (the number of spins), $k = 3$ (the number of successes), and $p=\frac{1}{3}$ (the probability of getting a vowel on a single spin). Then $1-p = 1-\frac{1}{3}=\frac{2}{3}$.

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

$C(5,3)=\frac{5!}{3!(5 - 3)!}=\frac{5!}{3!2!}=\frac{5\times4\times3!}{3!\times2\times1}=10$.

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

$P(X = 3)=C(5,3)\times(\frac{1}{3})^{3}\times(\frac{2}{3})^{2}=10\times\frac{1}{27}\times\frac{4}{9}=\frac{40}{243}\approx0.165$.

Answer:

$\frac{40}{243}$