Cards marked with numbers 1,3,5,..., 101 are placed in a bag and mixed thoroughly. A card is drawn at random from the bag. Find the probability that the number on the drawn card is (i) less than 19, (ii) a prime number less than 20.
Cards marked with numbers 1,3,5,..., 101 are placed in a bag and mixed thoroughly. A card is drawn at random from the bag. Find the probability that the number on the drawn card is (i) less than 19, (ii) a prime number less than 20.
Given number 1, 3, 5, . . . . . . . . , 101 form an AP with a = 1 and d = 2.
Let Tn = 101. Then,
1 + (n - 1)2 = 101
⇒ 1 + 2n - 2 = 101
⇒ 2n = 102
⇒ n = 51
Thus, total number of outcomes = 51.
(i) Let E1 be the event of getting a number less than 19.
Out of these numbers, numbers less than 19 are 1, 3, 5, ...... , 17.
Given number 1, 3, 5, ...... , 17 form an AP with a = 1 and d = 2.
Let Tn = 17. Then,
1 + (n - 1)2 = 17
⇒ 1 + 2n - 2 = 17
⇒ 2n = 18
⇒ n = 9
Thus, number of favorable outcomes = 9.
Therefore,
P(getting a number less than 19) = P(E1) = Number of outcomes favorable to E1 / Number of all possible outcomes
= 9/51= 3/17
Thus, the probability that the number on the drawn card is less than 19 is 3/17.
(ii) Let E2 be the event of getting a prime number less than 20.
Out of these numbers, prime numbers less than 20 are 3, 5, 7, 11, 13, 17 and 19.
Thus, the number of favorable outcomes = 7.
Therefore,
P(getting a prime number less than 20) = P(E2) = Number of outcomes favorable to E2/Number of all possible outcomes
= 7/51
Thus, the probability that the number on the drawn card is a prime number less than 20 is 7/51.
Given number 1, 3, 5, . . . . . . . . , 101 form an AP with a = 1 and d = 2.
Let Tn = 101. Then,
1 + (n - 1)2 = 101
⇒ 1 + 2n - 2 = 101
⇒ 2n = 102
⇒ n = 51
Thus, total number of outcomes = 51.
(i) Let E1 be the event of getting a number less than 19.
Out of these numbers, numbers less than 19 are 1, 3, 5, ...... , 17.
Given number 1, 3, 5, ...... , 17 form an AP with a = 1 and d = 2.
Let Tn = 17. Then,
1 + (n - 1)2 = 17
⇒ 1 + 2n - 2 = 17
⇒ 2n = 18
⇒ n = 9
Thus, number of favorable outcomes = 9.
Therefore,
P(getting a number less than 19) = P(E1) = Number of outcomes favorable to E1 / Number of all possible outcomes
= 9/51= 3/17
Thus, the probability that the number on the drawn card is less than 19 is 3/17.
(ii) Let E2 be the event of getting a prime number less than 20.
Out of these numbers, prime numbers less than 20 are 3, 5, 7, 11, 13, 17 and 19.
Thus, the number of favorable outcomes = 7.
Therefore,
P(getting a prime number less than 20) = P(E2) = Number of outcomes favorable to E2/Number of all possible outcomes
= 7/51
Thus, the probability that the number on the drawn card is a prime number less than 20 is 7/51.
