Question

The probability of the product of a perfect square when $2$ dice are thrown together is
1. $\dfrac{2}{9}$
2. $\dfrac{1}{9}$
3. $\dfrac{5}{{18}}$
4. None of these

Answer 1

Hint: In this question, we have to find the probability product of a perfect square. Here, two dice are thrown which means the number of possible outcomes will be $36$. Now from the sample space find those numbers whose product will be a perfect square and use the probability formula.

Formula Used:
$Probability = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}possible{\text{ }}outcomes}}$

Complete step by step Solution:
Given that,
Two dice are thrown together. Therefore, the sample space will be

$\left( {a,b} \right)$	$1$	$2$	$3$	$4$	$5$	$6$
$1$	$\left( {1,1} \right)$	$\left( {1,2} \right)$	$\left( {1,3} \right)$	$\left( {1,4} \right)$	$\left( {1,5} \right)$	$\left( {1,6} \right)$
$2$	$\left( {2,1} \right)$	$\left( {2,2} \right)$	$\left( {2,3} \right)$	$\left( {2,4} \right)$	$\left( {2,5} \right)$	$\left( {2,6} \right)$
$3$	$\left( {3,1} \right)$	$\left( {3,2} \right)$	$\left( {3,3} \right)$	$\left( {3,4} \right)$	$\left( {3,5} \right)$	$\left( {3,6} \right)$
$4$	$\left( {4,1} \right)$	$\left( {4,2} \right)$	$\left( {4,3} \right)$	$\left( {4,4} \right)$	$\left( {4,5} \right)$	$\left( {4,6} \right)$
$5$	$\left( {5,1} \right)$	$\left( {5,2} \right)$	$\left( {5,3} \right)$	$\left( {5,4} \right)$	$\left( {5,5} \right)$	$\left( {5,6} \right)$
$6$	$\left( {6,1} \right)$	$\left( {6,2} \right)$	$\left( {6,3} \right)$	$\left( {6,4} \right)$	$\left( {6,5} \right)$	$\left( {6,6} \right)$

Number of possible outcomes$ = 36$
Product of perfect squares are $\left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {4,1} \right),\left( {1,4} \right),\left( {3,3} \right),\left( {4,4} \right),\left( {5,5} \right),\left( {6,6} \right)} \right\}$
Number of favourable outcomes $ = 8$
Hence, $P\left( {Product{\text{ }}of{\text{ }}perfect{\text{ }}squares} \right) = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}possible{\text{ }}outcomes}}$
$P\left( {Product{\text{ }}of{\text{ }}perfect{\text{ }}squares} \right) = \dfrac{8}{{36}} = \dfrac{2}{9}$

Hence, the correct option is (1).

Note: In such type of problem, we must first determine the total number of possible outcomes (in this case, the total number of possible outcomes when two dice are thrown) and then determine the number of favourable outcomes (in this case, product of perfect square), after which we can directly apply the probability formula to determine the probability of receiving a favourable outcome, i.e., $Probability = \dfrac{{Number{\text{ }}of{\text{ }}favourable{\text{ }}outcomes}}{{Number{\text{ }}of{\text{ }}possible{\text{ }}outcomes}}$.