Question

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 of no correct answers.

Short answer

The probability of getting no correct answers is equal to 0.168.

Step-by-step solution

Given information

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.

Calculate the required probability

Let X denote 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 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 x=0.

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 no correct answer is computed below:

$$\begin{gathered} P\left(X=0\right)={}^8{C}_0{\left(0.20\right)}^0{\left(0.80\right)}^{8-0} \\ =\frac{8!}{0!\left(8-0\right)!}\times \left(0.20\right)\times {\left(0.80\right)}^8 \\ =0.167772 \\ \approx 0.168 \end{gathered}$$

Therefore, the probability of getting no correct answers is equal to 0.168.