Chapter 11, Problem 20CT

Solution

To calculate: The probability that the student guesses at least one of the three true and false questions correctly.

The probability that the student guesses at least one of the three true and false questions correctly is 0.875.

Given:

The total number of questions $n=3$

The probability that a student guesses an answer correctly is $p=\frac{1}{2}=0.5$

Formula used:

The probability for x success among n independent trials where the probability of each success is p is calculated using binomial probability distribution as shown below

  $\begin{array}{l}$ P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{(1-p)}^{n-x}$\left(\begin{array}{c}n\\ x\end{array}\right)=\frac{n!}{x!\left(n-x\right)!}\end{array}$

Calculation:

The probability that at least one of the questions is answered correctly can be calculated by subtracting the probability of no question is correct from 1 and the probability that no question is correct calculated as shown below

  $\begin{array}{l}$ P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{(1-p)}^{n-x}$$ P(X=0)=\left(\begin{array}{c}3\\ 0\end{array}\right){\left(\frac{1}{2}\right)}^0{\left(1-\frac{1}{2}\right)}^{3-0}$$ P(X=0)=\frac{3!}{0!\left(3-0\right)!}\times {0.5}^0\times {0.5}^3$P(X=0)=0.125\end{array}$

The probability that no questions is correct among three questions is 0.125.

Thus the probability that at least one question is correct among three questions is calculated as shown below

  $\begin{array}{c}P(atleastonequestioncorrect)=1-P(noquestioncorrrect)\\ =1-0.125\\ =0.875\end{array}$

Thus the probability that at least one of the question is guessed correctly is 0.875.