Question

20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the cards is:
(i) a multiple of 4?

(ii) not a multiple of 4?

(iii) odd?

(iv) greater than 12?

(v) divisible by 5?

(vi) not a multiple of 6?

Answer 1

We have 20 cards numbered from 1 to 20, one card is drawn at random

$\therefore n\left(s\right){=}^{20}{C}_1=20$

(i) Let E be the event that the number on the drawn cards is multiple of 4

$\therefore E=\{4,8,12,16,20\}$

$\therefore n\left(E\right)=5$

$\therefore P\left(E\right)=\frac{5}{20}=\frac{1}{4}$

(ii) Let E be the event that the number on the drawn card is not the multiple of 4

$\therefore \tilde{E}$ be the event that the number on the drawn card is the multiple of 4

$\therefore \tilde{E}\{4,8,12,16,20\}$

$\Rightarrow \tilde{E}=\frac{5}{20}=\frac{1}{4}$

$\therefore P\left(E\right)=1-\therefore P\tilde{E}$

$=1-\frac{1}{4}=\frac{3}{4}$

(iii) Let E be the event that the number of the drawn card is odd.

$\therefore E=\{1,3,5,7,9,11,13,15,17,19\}$

$\therefore n\left(E\right)=10$

$\therefore P\left(E\right)=\frac{10}{20}=\frac{1}{2}$

(iv) Let E be the event that number on the drawn card is greater that 12.

$\therefore E=\{13,14,15,16,17,18,19,20\}$

$\therefore n\left(E\right)=8$

$\Rightarrow P\left(E\right)=\frac{8}{20}=\frac{2}{5}$

(v) Let E be the event that number on the drawn card is divisible by 6.

$E=(5,10,15,20)$

$n\left(E\right)=4$

$\therefore P\left(E\right)=\frac{4}{20}=\frac{1}{5}$

(vi) Let E be the event that number on the drawn card is not divisible by 6.

$\therefore \tilde{E}$ be the event that number on the drawn card is divisible by 6

$\therefore \tilde{E}=\{6,12,18\}$

$\Rightarrow n\left(\tilde{E}\right)=3$

$\therefore P\left(\tilde{E}\right)=\frac{3}{20}$

$P\left(E\right)=1-P\left(\tilde{E}\right)$

$=1-\frac{3}{20}=\frac{17}{20}$