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. means \the probability of getting 3 of the spins.\ p(3 successes) p(4 successes) p(5 successes)

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 $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 5$, and from the first part, $p=\frac{1}{3}$, so $1-p=\frac{2}{3}$.

Step2: Calculate $C(n,k)$ for $k = 3$

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

Step3: Calculate $P(3\text{ successes})$

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

Answer:

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