Question

Cards marked with numbers 4 to 99 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is :
(i) A perfect square
(ii) A multiple of $7$
(iii) A prime number less than $30$
(iv) A perfect square between $91$ to $99$ .

Answer 1

Hint: In this question first we will find the total number of cards with numbers between \[4\] to $99$ . Now, we will find the perfect squares between \[4\] to $99$ After this we will find multiples of $7$ between \[4\] to $99$ . Now, find a prime number which is less than $30$ and at last we will find perfect squares between $91$ to $99$ . And now, for all the statements we will find the probability.

Complete step-by-step answer:
Number cards placed in a box marked with numbers \[4\] to $99$ are $99 - 4 + 1 = 96$ .
(i) A perfect square
Now, perfect square between \[4\] to $99$ are $4,\,9,\,16,\,25,\,36,\,49,\,64$ and $81$ . Therefore, the total number perfect
Squares are $8$ .
Therefore, the probability that the numbers on the card is a perfect square $ = \dfrac{8}{{96}} = \dfrac{1}{{12}}$ .
(ii) A multiple of $7$
The multiple of $7$ between \[4\] to $99$ are $7,\,14,\,21,\,28,\,35,\,42,\,49,\,56,\,63,\,70,\,77,\,84,\,91$ and $98$ . Therefore, the total number of multiples of $7$ are $14$ .
Therefore, the probability that the number on the card is a multiple of $7$ is $\dfrac{{14}}{{96}} = \dfrac{7}{{48}}$
(iii) ) A prime number less than $30$
The prime numbers less than $30$ are $5,\,7,\,11,\,13,\,17,\,19,\,23$ and $29$ . Therefore, the total number of prime numbers less than $30$ are $8$ .
Therefore, the probability that the number on the card is a prime number less than $30$ is $\dfrac{8}{{96}} = \dfrac{1}{{12}}$ .
(iv) ) A perfect square between $91$ to $99$ .
The perfect square between $91$ to $99$ is $0$ . Therefore, the probability that the number on the card is a perfect square between $91$ to $99$ is $\dfrac{0}{{96}} = 0$ .

Note: Students should be able to find the total numbers of cards. Students should also be aware about what is a perfect square and they also need to find out the prime numbers between some numbers.