Question

A number is randomly selected from the first 40 natural numbers. What is the probability that the selected number is divisible by 5 or 7?

• A. $\frac{8}{20}$
• B. $\frac{17}{40}$
• C. $\frac{7}{20}$
• D. $\frac{3}{10}$

Answer 1

The correct option is D $\frac{3}{10}$

The sample space for the given experiment is
S = {1, 2, 3, 4, ...., 39, 40}
n(S) = 40

Let A be the event that the number selected is divisible by 5.
Then A = {5, 10, 15, 20, 25, 30, 35, 40}
n(A) = 8
$\Rightarrow$P(A) = $\frac{8}{40}$

Let B be the event that the number selected is divisible by 7.
Then B = {7, 14, 21, 28, 35}
n(B) = 5
$\Rightarrow$P(B) = $\frac{5}{40}$

Now,
$A\cap B=\left\{35\right\}i.e.,n\left(A\cap B\right)=1\mathrm{Thus},P\left(A\cap B\right)=\frac{1}{40}$

By addition rule of probability,
$\begin{array}{ccc}P(A\cup B)& =& P\left(A\right)+P\left(B\right)-P(A\cap B)\\ & =& \frac{8}{40}+\frac{5}{40}-\frac{1}{40}\\ & =& \frac{12}{40}\\ & =& \frac{3}{10}\end{array}$
Therefore, probability that the selected number is divisible by 5 or 7 is $\frac{3}{10}.$