In Exercises 15–20, assume that random guesses are made for eight multiple choice questions on an SAT test, so that there are n = 8 trials, each with probability of success (correct) given by p = 0.20. Find the indicated probability for the number of correct answers.

Find the probability that the number x of correct answers is exactly 7.

A set of eight multiple-choice questions are answered in the SAT. The probability of a correct answer is given to be equal to 0.20.

Let X denote the number of correct answers.

Let success be defined as getting a correct answer.

Thus, the number of trials (n) is given to be equal to 8.

The probability of success (getting a correct answer) is p=0.20.

The probability of failure (getting a wrong answer) is obtained in the following manner:

$$\begin{gathered} q=1-p \\ =1-0.20 \\ =0.80 \end{gathered}$$

The number of successes required in eight trials is x=7.

By using the binomial probability formula, the probability of getting seven correct answers is computed in the following manner:

$$\begin{gathered} P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x} \\ P\left(X=7\right)={}^8{C}_7{\left(0.20\right)}^7{\left(0.80\right)}^1 \\ =\frac{8!}{7!1!}{\left(0.20\right)}^7{\left(0.80\right)}^1 \\ =0.00008192 \end{gathered}$$

Therefore, the probability of getting seven correct answers is equal to 0.00008192.