Tickets numbered 2, 3, 4, 5, ..... 100, 101 are placed in a box and mixed thoroughly. One ticket is drawn at random from the box. Find the probability that the number on the ticket is
(i) an even number
(ii) a number less than 16
(iii) a number which is a perfect square
(iv) a prime number less than 40.
Total numbers of elementary events are: 100
(i) Let E be the event of drawing even number ticket
The favourable numbers are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …..100
This forms an A.P where a= 2, d= 2 and an = 100
an= a + (n -1) d
100 = 2 + (n -1) 2
98/2 = n -1
49 + 1 = n
n = 50, being number of term
Then, the numbers of favourable outcomes = 50
∴ P (even number) = P (E) = 50/100 =1/2
(ii) Let E be the event of drawing number less than 16
The favourable numbers are: 2, 3, 4, 5, 6, …… 15
Then, the number of favourable outcomes = 14
∴ P (number > 6) = P (E)= 14/100 = 7/50
(iii) let E be the event of drawing a perfect square number
The favourable numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
The numbers of favourable outcomes = 10
∴ P (perfect square number) = P (E) = 10/ 100 = 1/10
(iv) let E be the event of drawing prime number less than 40
The favourable numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
Then, the numbers of favourable outcomes = 12
∴ P (prime number less than 40) = P (E) = 12/100 = 3/25