A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.
Solution:
There are a total of 90 discs, numbered from 1 to 90.
(i) Probability of getting a two-digit number:
The two-digit numbers range from 10 to 99.
So, the two-digit numbers in the box are 10, 11, 12, …, 90.
The total number of two-digit numbers is:
90 −10 + 1 = 81
Thus, the probability of drawing a two-digit number is:
(ii) Probability of getting a perfect square number:
Perfect square numbers are numbers that can be written as n2, where n is an integer. The perfect square numbers between 1 and 90 are:
12 = 1,22 = 4,32 = 9,42 = 16,52 = 25,62 = 36,72 = 49,82 = 64,92 = 81
So, the perfect square numbers in the box are 1, 4, 9, 16, 25, 36, 49, 64, 81, giving a total of 9 perfect square numbers.
Thus, the probability of drawing a perfect square number is:
(iii) Probability of getting a number divisible by 5:
The numbers divisible by 5 in the range from 1 to 90 are:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90
This gives a total of 18 numbers divisible by 5.
Thus, the probability of drawing a number divisible by 5 is:
