Problems on Probability

1. Experiment:
An operation which can produce some well-defined outcomes is called an experiment.
2. Random Experiment:
An experiment in which all possible outcomes are know and the exact output cannot be predicted in advance, is called a random experiment.
Examples:
1. Rolling an unbiased dice.
2. Tossing a fair coin.
3. Drawing a card from a pack of well-shuffled cards.
4. Picking up a ball of certain colour from a bag containing balls of different colours.
Details:
1. When we throw a coin, then either a Head (H) or a Tail (T) appears.
2. A dice is a solid cube, having 6 faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
3. A pack of cards has 52 cards.
It has 13 cards of each suit, name Spades, Clubs, Hearts and Diamonds.
Cards of spades and clubs are black cards.
Cards of hearts and diamonds are red cards.
There are 4 honours of each unit.
There are Kings, Queens and Jacks. These are all called face cards.
3. Sample Space:
When we perform an experiment, then the set S of all possible outcomes is called the sample space.
Examples:
1. In tossing a coin, S = {H, T}
2. If two coins are tossed, the S = {HH, HT, TH, TT}.
3. In rolling a dice, we have, S = {1, 2, 3, 4, 5, 6}.
4. Event:
Any subset of a sample space is called an event.
5. Probability of Occurrence of an Event:
Let S be the sample and let E be an event.
Then, E subset of S.
 P(E) = n(E) . n(S)
6. Results on Probability:
1. P(S) = 1
2. 0 <= P (E) <= 1
3. P() = 0
4. For any events A and B we have : P(A  B) = P(A) + P(B) - P(A  B)
5. If A denotes (not-A), then P(A) = 1 - P(A).
1.
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?
A.
 1 2
B.
 2 5
C.
 8 15
D.
 9 20
Explanation:
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
2.
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?
A.
 10 21
B.
 11 21
C.
 2 7
D.
 5 7
Explanation:
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
3.
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?
A.
 1 3
B.
 3 4
C.
 7 19
D.
 8 21
E.
 9 21
Explanation:
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
4.
What is the probability of getting a sum 9 from two throws of a dice?
A.
 1 6
B.
 1 8
C.
 1 9
D.
 1 12
Explanation:
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
5.
Three unbiased coins are tossed. What is the probability of getting at most two heads?
A.
 3 4
B.
 1 4
C.
 3 8
D.
 7 8
Explanation:
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
6.
Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?
A.
 1 2
B.
 3 4
C.
 3 8
D.
 5 16
Explanation:
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
7.
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:
A.
 21 46
B.
 25 117
C.
 1 50
D.
 3 25
Explanation:
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
8.
In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random. What is the probability of getting a prize?
A.
 1 10
B.
 2 5
C.
 2 7
D.
 5 7
Explanation:
 P (getting a prize) = 10 = 10 = 2 . (10 + 25) 35 7
9.
From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?
A.
 1 15
B.
 25 57
C.
 35 256
D.
 1 221
Explanation:
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
10.
Two dice are tossed. The probability that the total score is a prime number is:
A.
 1 6
B.
 5 12
C.
 1 2
D.
 7 9
Explanation:
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
11.
A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is:
A.
 1 13
B.
 2 13
C.
 1 26
D.
 1 52
Explanation:
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
12.
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:
A.
 1 22
B.
 3 22
C.
 2 91
D.
 2 77
Explanation:
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
13.
Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and one is a heart, is:
A.
 3 20
B.
 29 34
C.
 47 100
D.
 13 102
Explanation:
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
14.
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)?
A.
 1 13
B.
 3 13
C.
 1 4
D.
 9 52
Explanation:
Clearly, there are 52 cards, out of which there are 12 face cards.
 P (getting a face card) = 12 = 3 . 52 13
15.
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?
A.
 3 4
B.
 4 7
C.
 1 8
D.
 3 7