Probability

Total cards = 52. Fours = 4. Clubs = 13. The four of clubs is counted in both, so there are 4 + 13 - 1 = 16 cards that are either a four or a club. Cards that are neither = 52 - 16 = 36. Probability = 36/52 = 9/13.

Total outcomes = 6^3 = 216. Sum of 16 can be achieved by: (6,6,4) in 3 ways, (6,5,5) in 3 ways. Total favorable outcomes = 6. Probability = 6/216 = 1/36.

Let H be the number of heads and T be the number of tails. H + T = n and H - T = n - 3. Adding these, 2H = 2n - 3, so H = n - 1.5. Since H must be an integer, there is no possible outcome where the difference is n - 3. Thus, the probability is 0.

There are 36 total outcomes when two dice are thrown. The outcomes where both show the same number are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), which is 6 outcomes. Probability = 6/36 = 1/6.

P(Diamond) = 13/52. P(King) = 4/52. P(Diamond and King) = 1/52. P(Diamond or King) = P(Diamond) + P(King) - P(Diamond and King) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13.

When two dice are rolled, there are 36 possible outcomes. The pairs that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 favorable outcomes. Probability = 6/36 = 1/6.

Total balls = 15. Ways to choose 3 balls = 15C3 = (15*14*13)/(3*2*1) = 455. Ways to choose 1 red (from 9) and 2 white (from 6) = 9C1 * 6C2 = 9 * 15 = 135. Probability = 135/455 = 27/91.

Total coins = 15 + 25 + 10 = 50. One rupee coins = 15. Probability of selecting one rupee coin = 15/50 = 0.3. Probability of NOT selecting one rupee coin = 1 - 0.3 = 0.7.

The probability of getting exactly 7 heads when tossing 10 fair coins once is given by the binomial probability formula, which is 10C7 * (0.5)^10 = 120 / 1024. For 100 trials, the expected frequency is 100 * (120 / 1024), which is approximately 11.72. Rounding to the nearest whole number gives 12.

A regular hexagon inscribed in a circle has a side length equal to the radius r. The perimeter is 6 * r.

A standard deck has 52 cards, containing 4 kings and 4 queens. The probability of drawing a king or a queen is the sum of their individual probabilities, which is (4 + 4) / 52 = 8 / 52 = 2 / 13.

Prime sums greater than 3 are 5, 7, and 11. These occur in 4, 6, and 2 ways respectively, giving 12 favorable outcomes out of 36, or 1/3.

Initially there are 10 blue and 5 black balls (total 15). If one blue ball is drawn, 9 blue and 5 black balls remain (total 14). The probability that the second ball is blue is 9/14.