Probability problems
- 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:
Options- A.
1 22 - B.
3 22 - C.
2 91 - D.
2 77
Discuss
Correct Answer:
| 2 |
| 91 |
Explanation:
Let S be the sample space.
| Then, n(S) | = number of ways of drawing 3 balls out of 15 | |||
| = 15C3 | ||||
|
||||
| = 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 |
- 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?
Options- A.
1 2 - B.
2 5 - C.
8 15 - D.
9 20
Discuss
Correct Answer:
| 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 |
- 3. Two dice are tossed. The probability that the total score is a prime number is:
Options- A.
1 6 - B.
5 12 - C.
1 2 - D.
7 9
Discuss
Correct Answer:
| 5 |
| 12 |
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 |
- 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)?
Options- A.
1 13 - B.
3 13 - C.
1 4 - D.
9 52
Discuss
Correct Answer:
| 3 |
| 13 |
Explanation:
Clearly, there are 52 cards, out of which there are 12 face cards.
| ∴ P (getting a face card) = | 12 | = | 3 | . |
| 52 | 13 |
- 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:
Options- A.
3 20 - B.
29 34 - C.
47 100 - D.
13 102
Discuss
Correct Answer:
| 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 |
- 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:
Options- A.
1 13 - B.
2 13 - C.
1 26 - D.
1 52
Discuss
Correct Answer:
| 1 |
| 26 |
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 |
- 7. What is the probability of getting a sum 9 from two throws of a dice?
Options- A.
1 6 - B.
1 8 - C.
1 9 - D.
1 12
Discuss
Correct Answer:
| 1 |
| 9 |
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 |
- 8. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?
Options- A.
1 2 - B.
3 4 - C.
3 8 - D.
5 16
Discuss
Correct Answer:
| 3 |
| 4 |
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 |
- 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:
Options- A.
21 46 - B.
25 117 - C.
1 50 - D.
3 25
Discuss
Correct Answer:
| 21 |
| 46 |
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 ` | ||||
|
||||
| = 2300. |
| n(E) | = (10C1 x 15C2) | ||||||
|
|||||||
| = 1050. |
| ∴ P(E) = | n(E) | = | 1050 | = | 21 | . |
| n(S) | 2300 | 46 |
- 10. From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?
Options- A.
1 15 - B.
25 57 - C.
35 256 - D.
1 221
Discuss
Correct Answer:
| 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 |
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