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 at least one answer is correct.

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 Xdenote the number of correct answers.

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

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

The probability of failure (getting a wrong answer) is calculated below:

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

The number of successes required in eight trials should be at least equal to one.

The binomial probability formula is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

By using the binomial probability formula, the probability of getting at least one correct answer is computed below:

$$\begin{gathered} P\left(X⩾1\right)=1-P\left(X<1\right) \\ =1-P\left(X=0\right) \\ =1-{}^8{C}_0{\left(0.20\right)}^0{\left(0.80\right)}^8 \\ =1-0.168 \\ =0.832 \end{gathered}$$

Therefore, the probability of getting at least one correct answer is equal to 0.832.