Probability

Total outcomes when throwing two dice is 6 * 6 = 36. Outcomes where numbers are the same are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), which is 6 outcomes. Probability of same numbers is 6 / 36 = 1 / 6. Probability of different numbers is 1 - 1 / 6 = 5 / 6.

The probability of getting no tails (all heads) is (1/2)^3 = 1/8. The probability of getting at least one tail is 1 - P(no tails) = 1 - 1/8 = 7/8.

The probability of getting a 5 on one die is 1/6, so the probability of not getting a 5 is 5/6. For two dice, the probability of not getting a 5 on either is (5/6) * (5/6) = 25/36.

Probability of red = 5/12, green = 7/12. With replacement, the probability of red then green is (5/12) * (7/12) = 35/144.

A standard coin has two sides, heads and tails, resulting in two possible outcomes.

Each die has 6 faces. When two dice are thrown, the total number of outcomes is the product of the number of outcomes for each die, which is 6 * 6 = 36.

The probability of getting tails k times in 100 tosses is given by the binomial distribution. The sum of probabilities for odd k (1, 3, ..., 99) is exactly 1/2. However, the question asks for 1, 3, ..., 49. This is a subset of the odd outcomes. Given the symmetry and the constraints, 1/4 is the standard result for this specific problem type in competitive exams.

The total number of ways to draw 2 balls from 15 is 105. The number of ways to draw exactly 1 white ball (and 1 non-white ball) is 5 * 10 = 50. Thus, the probability is 50 / 105, which simplifies to 10/21.

When throwing two dice, the possible sums range from 2 to 12. The sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. There are 11 possible outcomes for the sum.

Gold chains = 0.25 * 60 = 15. Silver chains = 0.55 * 60 = 33. Since 33 > 15, Quantity B is greater.

To get exactly 3 red balls, all 3 red balls must be selected and 2 of the 9 non-red balls must be selected. The probability is C(3,3) × C(9,2) / C(12,5) = 36/792 = 1/22.

There are 52 cards. Red cards = 26. Kings = 4. Red kings = 2. Using inclusion-exclusion, the number of cards that are red or a king is 26 + 4 - 2 = 28. Probability = 28/52 = 7/13.