Chapter 11.9, Problem 40PPSE

Solution

To calculate: The grades which are considered to be rare in a test.

In the given test grades with 5 or 6 or 7 or 8 or 9 or 10 questions guessed being correct in the test are considered to be rare

Given:

The total number of questions in the test $n=10$

The number of options given for each question $=5$

Concept used:

The probability distribution function for a binomial random variable X with parameters n, and p is

  $P\left(X=x\right)=n_{C_x}{p}^x{(1-p)}^{n-x}$

Calculation:

From the given information there is only one correct answer per question, so the probability of guessing a question correct is $=\frac{1}{5}=0.2$$\begin{array}{l}n=10\\ p=0.2\end{array}$

The probability of guessing 10 questions correctly is calculated as shown below

  $\begin{array}{l}$ P(X=10)={10}_{C_{10}}{0.2}^{10}\times {(1-0.2)}^{10-10}$P(X=10)=1.02\times {10}^{-7}\end{array}$

The probability of guessing 9 questions correctly is calculated as shown below

  $\begin{array}{l}$ P(X=9)={10}_{C_9}{0.2}^9\times {(1-0.2)}^{10-9}$P(X=9)=4.096\times {10}^{-6}\end{array}$

The probability of guessing 8 questions correctly is calculated as shown below

  $\begin{array}{l}$ P(X=8)={10}_{C_8}{0.2}^8\times {(1-0.2)}^{10-8}$P(X=8)=7.37\times {10}^{-5}\end{array}$

The probability of guessing 7 questions correctly is calculated as shown below

  $\begin{array}{l}$ P(X=7)={10}_{C_7}{0.2}^7\times {(1-0.2)}^{10-7}$P(X=7)=0.00079\end{array}$

The probability of guessing 6 questions correct is calculated as shown below

  $\begin{array}{l}$ P(X=6)={10}_{C_6}{0.2}^6\times {(1-0.2)}^{10-6}$P(X=6)=0.0055\end{array}$

The probability of guessing 5 questions correct is calculated as shown below

  $\begin{array}{l}$ P(X=5)={10}_{C_5}{0.2}^5\times {(1-0.2)}^{10-5}$P(X=5)=0.02642\end{array}$

Interpretation:

The probabilities of exactly 5, 6, 7, 8, 9, 10 questions being correct when guessed are less than 0.05. Thus, all of these probabilities are considered to be rare. So, guessing 5 or more questions in the test is considered as a rare event.