Question 1

A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

Accepted Answer

Let S be the sample space. Then, n(S) = number of ways of drawing 3 balls out of 15 = 15C3 = (15 x 14 x 13) (3 x 2 x 1) = 455. Let E = event of getting all the 3 red balls. ∴ n(E) = 5C3 = 5C2 = (5 x 4) = 10. (2 x 1) ∴ P(E) = n(E) = 10 = 2 . n(S) 455 91

Question 2

Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

Accepted Answer

Here, S = {1, 2, 3, 4, ...., 19, 20}. Let E = event of getting a multiple of 3 or 5 = {3, 6 , 9, 12, 15, 18, 5, 10, 20}. ∴ P(E) = n(E) = 9 . n(S) 20

Question 3

Two dice are tossed. The probability that the total score is a prime number is:

Accepted Answer

Clearly, n(S) = (6 x 6) = 36. Let E = Event that the sum is a prime number. Then E = { (1, 1), (1, 2), (1, 4), (1, 6), (2, 1), (2, 3), (2, 5), (3, 2), (3, 4), (4, 1), (4, 3), (5, 2), (5, 6), (6, 1), (6, 5) } ∴ n(E) = 15. ∴ P(E) = n(E) = 15 = 5 . n(S) 36 12

Question 4

One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?

Accepted Answer

Clearly, there are 52 cards, out of which there are 12 face cards. ∴ P (getting a face card) = 12 = 3 . 52 13

Question 5

Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart, is:

Accepted Answer

Let S be the sample space. Then, n(S) = 52C2 = (52 x 51) = 1326. (2 x 1) Let E = event of getting 1 spade and 1 heart. ∴ n(E) = number of ways of choosing 1 spade out of 13 and 1 heart out of 13 = (13C1 x 13C1) = (13 x 13) = 169. ∴ P(E) = n(E) = 169 = 13 . n(S) 1326 102

Question 6

A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is:

Accepted Answer

Here, n(S) = 52. Let E = event of getting a queen of club or a king of heart. Then, n(E) = 2. ∴ P(E) = n(E) = 2 = 1 . n(S) 52 26

Question 7

What is the probability of getting a sum 9 from two throws of a dice?

Accepted Answer

In two throws of a dice, n(S) = (6 x 6) = 36. Let E = event of getting a sum ={(3, 6), (4, 5), (5, 4), (6, 3)}. ∴ P(E) = n(E) = 4 = 1 . n(S) 36 9

Question 8

Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

Accepted Answer

In a simultaneous throw of two dice, we have n(S) = (6 x 6) = 36. Then, E = {(1, 2), (1, 4), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (3, 4), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 2), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} ∴ n(E) = 27. ∴ P(E) = n(E) = 27 = 3 . n(S) 36 4

Question 9

In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:

Accepted Answer

Let S be the sample space and E be the event of selecting 1 girl and 2 boys. Then, n(S) = Number ways of selecting 3 students out of 25 = 25C3 ` = (25 x 24 x 23) (3 x 2 x 1) = 2300. n(E) = (10C1 x 15C2) = [ 10 x (15 x 14) ] (2 x 1) = 1050. ∴ P(E) = n(E) = 1050 = 21 . n(S) 2300 46

Question 10

From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?

Accepted Answer

Let S be the sample space. Then, n(S) = 52C2 = (52 x 51) = 1326. (2 x 1) Let E = event of getting 2 kings out of 4. ∴ n(E) = 4C2 = (4 x 3) = 6. (2 x 1) ∴ P(E) = n(E) = 6 = 1 . n(S) 1326 221

Question 11

A bag contains 6 black and 8 white balls. One ball is drawn at random. What is the probability that the ball drawn is white?

Accepted Answer

Let number of balls = (6 + 8) = 14. Number of white balls = 8. P (drawing a white ball) = 8 = 4 . 14 7

Question 12

In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

Accepted Answer

Total number of balls = (8 + 7 + 6) = 21. Let E = event that the ball drawn is neither red nor green = event that the ball drawn is blue. ∴ n(E) = 7. ∴ P(E) = n(E) = 7 = 1 . n(S) 21 3

Question 13

Three unbiased coins are tossed. What is the probability of getting at most two heads?

Accepted Answer

Here S = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH} Let E = event of getting at most two heads. Then E = {TTT, TTH, THT, HTT, THH, HTH, HHT}. ∴ P(E) = n(E) = 7 . n(S) 8

Question 14

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

Accepted Answer

Total number of balls = (2 + 3 + 2) = 7. Let S be the sample space. Then, n(S) = Number of ways of drawing 2 balls out of 7 = 7C2 ` = (7 x 6) (2 x 1) = 21. Let E = Event of drawing 2 balls, none of which is blue. ∴ n(E) = Number of ways of drawing 2 balls out of (2 + 3) balls. = 5C2 = (5 x 4) (2 x 1) = 10. ∴ P(E) = n(E) = 10 . n(S) 21

Question 15

In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?

Accepted Answer

P (getting a prize) = 10 = 10 = 2 . (10 + 25) 35 7

Question 16

What is the probability that a number selected from the numbers 1, 2, 3, 4, 5....16 is a prime number?

Accepted Answer

S = {1, 2, 3, ......16} E = {2, 3, 5, 7, 11, 13} ∴ P(E) = n(E)/n(S) = 6/16= 3/8

Question 17

An urn contains 9 red, 7 white and 4 black balls. A ball is drawn at random. What is the probability that the ball drawn is not red?

Accepted Answer

P (red) = 9 / (9 + 7 + 4) = 9/20 ∴ P(not-red) = (1 - 9/20) = 11/20

Question 18

In a lottery there are 20 prizes and 15 blanks. What is the probability of getting prize?

Accepted Answer

P (getting a prize ) = 20 / (20 + 15 ) = 20 / 35 = 4/7

Question 19

The odds in favour of an event are 3 : 5. The probability of occurrence of the event is?

Accepted Answer

Number of cases favourable of E = 3 Total Number of cases = (3 + 5 ) = 8 ∴ P(E) = 3/8

Question 20

The probability of getting heads in both trials when a balanced coin is tossed twice will be?

Accepted Answer

Probability of getting head in one trail = 1/2 ∴ Reqd Probability of getting heads in both the trails = 1/2 x 1/2 = 1/4

Probability Questions