Question

A box contains 20 cards marked with the numbers 1 to 20. One card is drawn from the box at random. What is the probability that number on the card is
(i) a prime number
(ii) perfect square
(iii) multiple of 5.

Answer 1

Sample space, S = {1, 2, 3, 4,..., 20}
∴ n(S) = 20

(i) Let A be the event that the number on the card is a prime number.
We have:
A = {2, 3, 5, 7, 11, 13, 17, 19}
∴ n(A) = 8
Thus, we get:
$$\begin{gathered} \mathrm{P}\left(\mathrm{A}\right)=\frac{n\left(\mathrm{A}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{8}{20}=\frac{2}{5} \end{gathered}$$

(ii) Let B be the event that the number on the card is a perfect square.
We have:
B = {1, 4, 9, 16}
∴ n(B) = 4
Thus, we get:
$$\begin{gathered} \mathrm{P}\left(\mathrm{B}\right)=\frac{n\left(\mathrm{B}\right)}{n\left(\mathrm{B}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{B}\right)=\frac{4}{20}=\frac{1}{5} \end{gathered}$$
(iii) Let C be the event that the number on the card is a multiple of 5.
We have:
C = {5, 10, 15, 20}
∴ n(C) = 4
Thus, we get:
$$\begin{gathered} \mathrm{P}\left(\mathrm{C}\right)=\frac{n\left(\mathrm{C}\right)}{n\left(\mathrm{S}\right)} \\ \Rightarrow \mathrm{P}\left(\mathrm{C}\right)=\frac{4}{20}=\frac{1}{5} \end{gathered}$$