Chapter 11.3, Problem 32PPSE

Solution

To identify:The probability that you will answer all three correctlyA multiple choice test has four choices for each answer. Suppose you make a random guess on three of the ten test questions.

*Is each guess a dependent event or an independent event?

*What is the probability that a random guess on one question will yield thecoorect answer?

The probability $P\left(multipleof4and5\right)=40\%$

Given information:

A multiple choice test has four choices for each answer. Suppose you make a random guess on three of the ten test questions

Explanation:

Consider the given information.

Determine the probability that a multiple of 4 or 5 is chosen from 1 to 100

Multiple of 4 from 1 to 100

Since here last term of the series is 100 so by using formula of nth term of AP.

  $\begin{array}{l}{a}_n=a\left(n-1\right)d\\ 100=4+\left(n-1\right)4\\ 100-4=\left(n-1\right)4\\ 96=\left(n-1\right)4\\ \frac{96}{4}=\left(n-1\right)\\ 24=\left(n-1\right)\\ n=24+1\\ n=25\end{array}$

So, number of multiple of 4 is 25

  $P\left(multipleof4\right)=\frac{1}{4}$

Multiple of 5 from 1 to 100

  $5,10,15,\mathrm{20.....100}$

Since here last term of these series is 100 so by using formula of nth term of AP.

  $\begin{array}{l}{a}_n=a\left(n-1\right)d\\ 100=5+\left(n-1\right)5\\ 100-5=\left(n-1\right)5\\ 95=\left(n-1\right)5\\ \frac{95}{5}=\left(n-1\right)\\ 19=\left(n-1\right)\\ n=19+1\\ n=20\end{array}$

So, number of multiple of 5 is 20

Therefore

  $P\left(multipleof5\right)=\frac{1}{5}$

Number which are multiple of both 4 and 5 are

20,40,60,80,100

So, there are 5 number that are divisible by both 4 and 5

So

  $\begin{array}{l}P\left(multipleof4and5\right)=\frac{5}{100}\\ P\left(multipleof4and5\right)=\frac{1}{20}\end{array}$

Therefore

  $P\left(multipleof4and5\right)+P\left(multipleof5\right)-\left(multipleof4and5\right)$

  $\begin{array}{l}P\left(multipleof4and5\right)+P\left(multipleof5\right)-\left(multipleof4and5\right)\\ \frac{25}{100}+\frac{20}{100}-\frac{5}{100}\\ \frac{40}{\begin{array}{l}100\\ \frac{2}{5}\\ 40\%\end{array}}\end{array}$

Hence $P\left(multipleof4and5\right)=40\%$