At Least One. In Exercises 5–12, find the probability.

At Least One Correct Answer If you make random guesses for 10 multiple choice SAT test questions (each with five possible answers), what is the probability of getting at least 1 correct? If these questions are part of a practice test and an instructor says that you must get at least one correct answer before continuing, is there a good chance you will continue?

Ten multiple-choice questions need to be answered in an SAT.

Each question has five possible answers.

The probability of an event occurring “at least once” (once or more) is complementary to the probability that the event does not occur.

$P\left(\text{occurrence}\text{of}\text{A}\text{at}\text{least}\text{once}\right)=1-P\left(\text{occurrence}\text{of}\text{not}\text{A}\right)$

Let A be the event of answering a question correctly.

The total number of choices for one question is five.

The number of correct choices for any given question is one.

The probability of answering one question correctly is:

$$\begin{gathered} P\left(A\right)=\frac{1}{5} \\ =0.2 \end{gathered}$$

Thus, $P\left(A\right)$is 0.2.

The probability of answering a question incorrectly is:

$$\begin{gathered} P\left(\overline{A}\right)=1-P\left(A\right) \\ =1-0.2 \\ =0.8 \end{gathered}$$

Thus, $P\left(\overline{A}\right)$is equal to 0.8.

Ten questions are answered.

The probability of not answering any of the 10 questions correctly is given by:

$$\begin{gathered} P\left(\text{all}\text{incorrect}\right)=\left(0.8\right)\times \left(0.8\right)\times ...\times \left(0.8\right) \\ ={\left(0.8\right)}^{10} \\ =0.107 \end{gathered}$$

The probability of answering at least one question correctly out of 10 is computed as follows:

$$\begin{gathered} P\left(\text{at}\text{least}\text{1}\text{correct}\text{answer}\right)=1-P\left(\text{all}\text{incorrect}\right) \\ =1-0.107 \\ =0.893 \end{gathered}$$

Therefore, the probability of answering at least one question correctly is equal to 0.893.

The probability value of answering at least one question correctly is high(approximately 0.9).

Thus, there is a fair chance of continuing with the practice test.