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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
a. $\dfrac{48}{113},\dfrac{16}{113}$
b. $\dfrac{8}{113},\dfrac{16}{113}$
c. $\dfrac{8}{113},\dfrac{16}{143}$
d. $\dfrac{8}{113},\dfrac{16}{155}$

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Last updated date: 29th Mar 2024
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MVSAT 2024
Answer
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Hint:In order to solve this question, we should know about the concept of probability, that is, probability is the ratio of favourable outcomes to the total number of outcomes. Or we can say that,
$\text{probability=}\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$. So, we will find the total outcomes first and then the favourable outcomes for each case and then we will be able to find the answer.

Complete step-by-step answer:
In this question, we have been asked to find the probability of obtaining a square number and a multiple of 7, when a card has been drawn from a box containing cards numbered 11 to 123.
Now, we know that the cards are numbered from 11 to 123. So, we can say that the number of cards in the box are 123 – 10 = 113. So, we can say that the total number of outcomes in each case is 113.
Now, we know that probability is calculated using the formula, that is, $\text{probability=}\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$. So, we will calculate the favourable outcomes for each case.
(i) A square number
Now, we know that from 11 to 123, there are 16, 25, 36, 49, 64, 81, 100 and 121 as the only perfect squares. So, we can say that there are 8 square numbers from 11 to 123, that is, favourable outcomes are 8. Therefore, we can say,
\[\text{probability of a square number =}\dfrac{8}{113}\]
(i) A multiple of 7
Now, we know that from 11 to 123, there are 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112 and 119 as the multiples of 7. So, we can say that there are 16 multiples of 7. Therefore, we can say that the favourable outcomes are 16. Hence, we get,
$\text{probability of getting a multiple of 7 =}\dfrac{16}{113}$
From the above observations, we can say that $\dfrac{8}{113},\dfrac{16}{113}$ are the probabilities of getting a square number and a multiple of 7 respectively when a card is chosen at random from a box of cards numbered 11 to 123. Hence, option (b) is the correct answer.

Note: While solving this question, the possible mistake one can make is at the time of choosing the option. In a hurry, we might choose option (c) or (d) as the correct answer because of the numerators in both the probability, but the denominators in both the cases are wrong. So, we have to be very patient and careful while selecting the correct answer. Also, while taking the total number of cards in the box, one might write it as, 123 – 11 = 112, but this is wrong, we have to actually write it as, 123 – 10 = 113.