Question

A bag contains tickets numbered from I to 20. Two tickets are drawn. Find the probability that

(i) both the tickets have prime numbers on them

(ii) on one there is a prime number and on the other there is a multiple of 4.

Answer 1

Clearly, the sample space is given by
$S=\{1,2.3,4,5,\dots ,19,20\}$

$\therefore n\left(S\right)=20C_2=190$

(i) Let $E_1$ be the event where both the tickets have prime numbers on them

Then $E_1,=\{2,3,5,7,11,13,17,19\}$

$\therefore$ Favourable number ofways = $n\left(E_1\right){=}^8{C}_2$

Hence, required probability = $P\left(E_1\right)$

$=\frac{n\left(E_1\right)}{n\left(S\right)}=\frac{{}^8{C}_2}{{}^{20}{C}_2}=\frac{14}{95}$

Let $E_2$ be the event where one ticket has a prime number, while the other has a multiple of 4.

Then prime number = {2, 3, 5, 7, 11, 13, 17,19}
and multiples of 4 = {4, 8,12, 16, 20}

$\therefore$ Favourable number of ways, $n\left(E_2\right){=}^8{C}_1{\times }^5{C}_1=8\times 5=40$

Hence, required probability, $P\left(E_2\right)$

$=\frac{n\left(E_2\right)}{n\left(S\right)}=\frac{40}{190}=\frac{4}{19}$