Chapter 11.4, Problem 5CYU

To determine

Write the correct option.

Answer to Problem 5CYU

  $\boxed{D}$ is correct option.

Explanation of Solution

Given information:

Aiden has earned five spines of the wheel on the right. He will receive a prize each time the spinner lands on WIN. What is the probability that he receives three prizes?

A $4.2\%$ C $7.1\%$

B $5.8\%$ D $8.8\%$

Calculation:

Consider a spinner divided into $8$ equal parts which is given into the problem.

Generally, the probability of event $E$ is,

  $P(E)=\frac{Number-of-favorable-outcomes}{Total-number-of-outcomes}$

In the above spinner, $2$ out of $8$ parts has WIN written on it. So, the number of favorable outcomes is $2$ while the total number of outcomes is $8$ . So, the probability of success or the probability that the spinner lands on WIN is,

  $\begin{array}{l}p=\frac{2}{8}\\ =\frac{1}{4}\\ =0.25\end{array}$

The probability failure is ,

  $\begin{array}{l}q=1-p\\ =1-0.25\\ =0.75\end{array}$

The probability of $k$ successes in $n$ trials of a binomial experiment is :

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure.

Substitute $p=0.25,q=0.75,k=3,n=5$ in the above formula to find the probability of the spinner laanding on WIN in $3$ out of $5$ trials,

  $\begin{array}{l}$ P(3){=}^5{C}_3{(0.25)}^3{(0.75)}^{5-3}$$ {=}^5{C}_3{(0.25)}^3{(0.75)}^2$$ =10(0.015625)(0.2625)$=0.0879\\ P(3)\approx 8.8\%\end{array}$

So , the probability that the spinner lands on WIN in $3$ out $5$ spins is $8.8$ .

Hence, option $\boxed{D}$ is correct.