Question

Two different dice are thrown together. Find the probability that the product of the number appeared is less than 18.

OR

Harpreet tosses two different coins simultaneously. What is the probability that she gets:
(i) at least one head ?
(ii) one head and one tail ?

Answer 1

There are $6\times 6=36$ possible outcomes.

n(S)=36

Favourable outcomes are $(4,2),(4,3),(4,5),(5$, 1), $(5,2),(5,3),(5,3),(6,1),(6,2),(1,1),(1,2)$, $(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2$, $6),(3,1),(3,2),(3,3),(3,4),(2,5)$ and $(4,1)$.
No. of favourable outcomes,
$$\begin{array}{cc}& n\left(E\right)=26\\ & P\text{(Product appears is less than 18)}\\ & P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{26}{36}=\frac{13}{18}\end{array}$$

OR

All possible outcomes are $\{HH,TT,TH,HT\}$

n(S)=4

(i) At least one head,
All favourable outcome are $\{HH,TH,HT\}$
$$\begin{array}{cc}& n\left(E_1\right)=3\\ & P\left(E_1\right)=\frac{n\left(E_1\right)}{n\left(S\right)}=\frac{3}{4}\end{array}$$
(ii) One head and one tail
All favourable outcome are $\{TH,HT\}$
$$\begin{array}{cc}n\left(E_2\right)& =2\\ P\left(E_2\right)& =\frac{n\left(E_2\right)}{n\left(S\right)}\\ & =\frac{2}{4}=\frac{1}{2}\end{array}$$