Probability

Total ways to choose 5 tickets from 30 is 30C5. For the 3rd number to be 20, we must choose 2 numbers from {1, ..., 19} and 2 numbers from {21, ..., 30}. Ways = 19C2 * 10C2 = (19*18/2) * (10*9/2) = 171 * 45 = 7695. Total ways = 30C5 = 142506. Probability = 7695 / 142506 = 285 / 5278.

Multiples of 3 between 1 and 20 are: 3, 6, 9, 12, 15, 18. There are 6 such numbers. Probability = 6/20 = 3/10.

Primes between 1 and 20: 2, 3, 5, 7, 11, 13, 17, 19. There are 8 primes. Probability = 8/20 = 2/5.

Total outcomes for 3 coin tosses = 2^3 = 8. The only outcome with 3 heads is (H, H, H). Probability = 1/8.

When tossing three coins, there are 2^3 = 8 possible outcomes. The only outcome with no heads is (T, T, T), which occurs once. Thus, the probability is 1/8.

When tossing three coins, there are 2^3 = 8 possible outcomes. Only one outcome (T, T, T) consists of 3 tails, so the probability is 1/8.

Total outcomes for 3 coins = 2^3 = 8. Only one outcome has no tails (HHH). Probability = 1/8.

Total balls = 9 + 7 + 4 = 20. Red balls = 9. Probability = 9/20.

Probability of rolling a 6 is 1/6, not rolling is 5/6. A wins on 1st, 3rd, 5th... turn. P(A wins) = 1/6 + (5/6)(5/6)(1/6) + ... = (1/6) / (1 - 25/36) = (1/6) / (11/36) = 6/11.

Sum of two dice = 10: (4,6), (5,5), (6,4) -> 3 outcomes. P(win) = 3/36 = 1/12. P(lose) = 11/12. A wins on 1st, 3rd, 5th... turn. P(A) = 1/12 + (11/12)^2 * 1/12 + ... = (1/12) / (1 - (11/12)^2) = (1/12) / (1 - 121/144) = (1/12) / (23/144) = 12/23.

Total discs = 9. The probability of drawing 3 blue discs without replacement is (4/9) * (3/8) * (2/7) = 24 / 504. Simplifying this fraction, 24 / 504 = 1 / 21.

Total balls = 5 + 7 = 12. Probability of white = (Number of white) / (Total) = 5 / 12.

This is a binomial distribution problem with n=9, p=0.5. Probability of at least 5 heads is P(X>=5). Since the distribution is symmetric, P(X>=5) = P(X<=4). P(X>=5) + P(X<=4) = 1. 2*P(X>=5) = 1, so P(X>=5) = 0.5.

Total balls = 12. Ways to draw 2 = 12C2 = 66. Ways to draw 2 green = 7C2 = 21. Ways to draw 2 yellow = 5C2 = 10. Total favorable = 21+10 = 31. Probability = 31/66.