Question

A box contains cards numbered 11 to 123. A card is drawn at random from the box, Find the probability that the number on the drawn card is
I.A square number
II.A multiple of 7

Answer 1

Hint : Here the given question is based on the concept of probability. We have to find the probability of choosing a square number card and cards which have multiple of 7, by using the definition of probability and on further simplification we get the required probability of choosing a card.

Complete step-by-step answer :
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
$Probability \;of \;event \;to \;happen P\left( E \right) = \dfrac{{Number\; of \;favourable outcomes}}{{Total\; Number\; of\; outcomes}}$
Consider the given question:
A box contains cards numbered 11 to 123. Cards are drawn randomly from the box,
Total cards numbered from $11$ to 123 = 113
now we have to find
I.Probability of drawn cards are square numbers.
The square numbers from 11 to 123 are 16, 25, 36, 49, 64, 81, 100 and 121
Total square number from 11 to 123 = 8
By the definition of probability
$\Rightarrow \,\,P\left( {square{\text{ }}number} \right) = \dfrac{{Total{\text{ }}square{\text{ }}numbes}}{{Total{\text{ }}numbered{\text{ }}cards{\text{ }}from{\text{ }}11{\text{ }}\;to{\text{ }}123}}$
$\Rightarrow \,\,P\left( {square\,number} \right) = \dfrac{8}{{113}}$

II.Probability of drawn card are multiple of 7
Multiples of 7 from 11 to 123 are
14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112 and 119
Total multiples of 7 from 11 to 123 = 16
By the definition of probability
$\Rightarrow \,\,P\left( {multiples\,of\,7} \right) = \dfrac{{Total{\text{ }}multiple{\text{ }}numbes{\text{ of }}7}}{{Total{\text{ }}numbered{\text{ }}cards{\text{ }}from{\text{ }}1{\text{ }}\;to{\text{ }}123}}$
$\Rightarrow \,\,P\left( {multiples\,of\,7} \right) = \dfrac{{16}}{{113}}$
Hence, it’s a required solution.

Note : The probability is a number of possible values. Candidate must know we have to use the permutation concept or combination concept to solve the given problem because it is the first and main thing to solve the problem. Here we arrange the things in the possible ways so we are using the concept permutation.