Chapter 6, Problem 12T

Solution

To Calculate:

The probability of randomly guessing at least $8$ correct answers on a $10$ question true-or-false quiz.

  $\frac{7}{128}$

Given:

There are $10$ true-or-false type question in the quiz.

Concepts Used:

The probability to correctly answer a true or false type question at random is $\frac{1}{2}$ .

The correctness of the answer to each question is independent of the correctness of the answers to the other questions.

In a binomial experiment with $n$ independent trials each having probability of success $p$ . The probability of $r$ successes is given by ${}_n{C}_r{p}^r{\left(1-p\right)}^{n-r}$ .

The probability of occurrence of the union of many events that are mutually exclusive is the sum of their individual probabilities.

The definition ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$ .

Calculations:

Calculate the probability of randomly guessing at least $8$ correct answers on a $10$ question true-or-false quiz. The experiment is equivalent to a binomial experiment with $10$ independent trials each having probability of success $\frac{1}{2}$ .

  $\begin{array}{c}P\left(\text{at least 8 correct}\right)=P\left(\text{8 correct}\right)+P\left(\text{9 correct}\right)+P\left(\text{10 correct}\right)\\ {=}_{10}{C}_8{\left(\frac{1}{2}\right)}^8{\left(1-\frac{1}{2}\right)}^{10-8}{+}_{10}{C}_9{\left(\frac{1}{2}\right)}^9{\left(1-\frac{1}{2}\right)}^{10-9}{+}_{10}{C}_{10}{\left(\frac{1}{2}\right)}^{10}{\left(1-\frac{1}{2}\right)}^{10-10}\\ =\frac{10\times 9}{2}{\left(\frac{1}{2}\right)}^{10}+10{\left(\frac{1}{2}\right)}^{10}+{\left(\frac{1}{2}\right)}^{10}\\ ={\left(\frac{1}{2}\right)}^{10}\left(45+10+1\right)\\ =\frac{7}{128}\end{array}$

Conclusion:

  $\frac{7}{128}$ is the probability of randomly guessing at least $8$ correct answers on a $10$ question true-or-false quiz.