Question 1

A coin is tossed three times. What is the probability of getting at least one head?

Accepted Answer

Step 1: Total possible outcomes when tossing a coin 3 times = 2³ = 8. Step 2: Outcomes with at least one head = Total outcomes - Outcomes with no heads (all tails). Step 3: Outcomes with all tails = 1 (TTT). Step 4: Outcomes with at least one head = 8 - 1 = 7. Step 5: Probability = 7/8. Therefore, the answer is 7/8.

Question 2

A bag contains 5 red balls, 7 blue balls, and 8 green balls. If two balls are drawn simultaneously at random, what is the probability that both balls are of the same colour?

Accepted Answer

Step 1: Total balls = 5 + 7 + 8 = 20. Total ways to draw 2 balls = C(20,2) = 190. Step 2: Ways to draw 2 red balls = C(5,2) = 10. Ways to draw 2 blue balls = C(7,2) = 21. Ways to draw 2 green balls = C(8,2) = 28. Step 3: Favourable outcomes = 10 + 21 + 28 = 59. Step 4: Probability = 59/190.

Question 3

In a lottery, the probability of winning a prize is 0.12. If 500 tickets are sold, how many tickets are expected to win a prize?

Accepted Answer

Step 1: Probability of winning = 0.12. Step 2: Number of tickets sold = 500. Step 3: Expected number of winning tickets = Probability × Total tickets = 0.12 × 500 = 60.

Question 4

A student has a 60% chance of passing Mathematics and a 75% chance of passing English. Assuming independence, what is the probability that the student passes at least one subject?

Accepted Answer

Step 1: P(Math) = 0.60, P(English) = 0.75. Step 2: Use complement: P(at least one pass) = 1 - P(both fail). Step 3: P(fail Math) = 1 - 0.60 = 0.40. Step 4: P(fail English) = 1 - 0.75 = 0.25. Step 5: P(both fail) = 0.40 × 0.25 = 0.10. Step 6: P(at least one pass) = 1 - 0.10 = 0.90.

Question 5

Three cards are drawn from a standard deck of 52 cards without replacement. What is the probability that all three cards are of the same suit?

Accepted Answer

Step 1: Total ways to draw 3 cards from 52 = C(52,3) = 22,100. Step 2: There are 4 suits, each with 13 cards. Step 3: Ways to draw 3 cards from one suit = C(13,3) = 286. Step 4: Ways to draw 3 cards all from same suit = 4 × C(13,3) = 4 × 286 = 1,144. Step 5: Probability = 1,144/22,100 = 286/5,525 = 22/425.

Question 6

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, what is the probability of drawing either a red or a green ball?

Accepted Answer

Step 1: Total number of balls = 5 + 3 + 2 = 10. Step 2: Number of red balls = 5, number of green balls = 2. Step 3: Number of favorable outcomes (red or green) = 5 + 2 = 7. Step 4: Probability = Favorable outcomes / Total outcomes = 7/10 = 0.7. Therefore, the answer is 7/10.

Question 7

A number is selected at random from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. What is the probability that the number is either even or a multiple of 3?

Accepted Answer

Step 1: Total numbers = 10. Step 2: Even numbers: {2, 4, 6, 8, 10} = 5 numbers. Step 3: Multiples of 3: {3, 6, 9} = 3 numbers. Step 4: Numbers that are both even and multiples of 3: {6} = 1 number. Step 5: Using inclusion-exclusion: Even or multiple of 3 = 5 + 3 - 1 = 7 numbers {2, 3, 4, 6, 8, 9, 10}. Step 6: Probability = 7/10. Therefore, the probability is 7/10.

Question 8

A bag contains 5 red balls, 7 blue balls, and 8 green balls. If one ball is drawn at random, what is the probability that it is either red or green?

Accepted Answer

Step 1: Total number of balls = 5 + 7 + 8 = 20. Step 2: Number of red or green balls = 5 + 8 = 13. Step 3: Probability = 13/20. Therefore, the answer is 13/20.

Question 9

A bag contains 5 red balls, 7 blue balls, and 8 green balls. If two balls are drawn simultaneously at random, what is the probability that both balls are of the same colour?

Accepted Answer

Step 1: Total balls = 5 + 7 + 8 = 20. Total ways to draw 2 balls = C(20,2) = 190. Step 2: Ways to draw 2 red balls = C(5,2) = 10. Ways to draw 2 blue balls = C(7,2) = 21. Ways to draw 2 green balls = C(8,2) = 28. Step 3: Total favourable outcomes = 10 + 21 + 28 = 59. Step 4: Probability = 59/190.

Question 10

In a lottery, 6 numbers are drawn from 1 to 49. A player buys one ticket with 6 numbers. What is the probability that the player matches exactly 4 of the 6 drawn numbers?

Accepted Answer

Step 1: Total ways to choose 6 from 49 = C(49,6) = 10,068,347. Step 2: To match exactly 4: choose 4 from the 6 winning numbers AND 2 from the 43 non-winning numbers. Ways = C(6,4) × C(43,2) = 15 × 903 = 13,545. Step 3: Probability = 13,545 / 10,068,347 ≈ 1/743.

Question 11

A box contains 3 defective items and 12 non-defective items. If 3 items are drawn at random without replacement, what is the probability that exactly 2 items are defective?

Accepted Answer

Step 1: Total items = 3 + 12 = 15. Total ways to choose 3 items = C(15,3) = 455. Step 2: Ways to choose exactly 2 defective items = C(3,2) × C(12,1) = 3 × 12 = 36. Step 3: Probability = 36/455. Step 4: Simplify: GCD(36, 455) = 1, so 36/455 ≈ 0.0791.

Question 12

Three unbiased coins are tossed simultaneously. Given that at least one head appears, what is the conditional probability that exactly two heads appear?

Accepted Answer

Step 1: Sample space with at least one head = {HHH, HHT, HTH, HTT, THH, THT, TTH} = 7 outcomes. Step 2: Outcomes with exactly two heads = {HHT, HTH, THH} = 3 outcomes. Step 3: Conditional probability = 3/7.

Question 13

A bag contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn simultaneously without replacement. What is the probability that both balls are of different colours?

Accepted Answer

Step 1: Total balls = 5 + 7 + 8 = 20. Total ways to draw 2 balls = C(20,2) = 190. Step 2: Ways to draw both same colour = C(5,2) + C(7,2) + C(8,2) = 10 + 21 + 28 = 59. Step 3: Ways to draw different colours = 190 - 59 = 131. Step 4: Probability = 131/190.

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