Question

If the number x is chosen from the numbers 1,2,3 and a number y is selected from the number 1,4,9 find the probability that xy<9.

Answer 1

Hint- We are asked to find probability which in turn comes with the demand of framing sample spaces. So first we will find out the number of ways in which x can be selected and then the number of ways in which y can be selected so as to find out all the possibilities. Then using the condition XY<9 find out how many possibilities we are left with and then proceed.

Complete step-by-step answer:

Number ‘x’ can be selected in the three ways and corresponding to each such way there are three ways of selecting number y therefore, the two numbers can be selected in 9 ways as listed below:
$\left( {1,1} \right),\left( {1,4} \right),\left( {1,9} \right),\left( {2,1} \right),\left( {2,4} \right),\left( {2,9} \right),\left( {3,1} \right),\left( {3,4} \right),\left( {3,9} \right)$
Total number of outcomes = 9
The product xy will be less than 9, if x and y are chosen in one of the following ways:
$\left( {1,1} \right),\left( {1,4} \right),\left( {2,1} \right),\left( {2,4} \right),\left( {3,1} \right)$
Number of favorable outcomes = 5

Using formula, probability = (Number of favourable outcomes/Total number of possible outcomes)

Probability (product less than 9) = $\dfrac{5}{9}$

Note: For solving these types of questions you must be able to frame the sample spaces. Sample space in probability(also called as sample description space or possibility space) of an experiment or a random trial is the set of all possible outcomes or results of that experiment.