Probability

Each coin has 2 outcomes (H or T). For 2 coins, total outcomes = 2 * 2 = 4 (HH, HT, TH, TT).

A coin has two sides, heads and tails. Assuming a fair coin, the probability of any specific side landing up is 1 divided by 2.

Two events are mutually exclusive if they cannot occur at the same time. C (sum < 4) includes sums {2, 3}, while D (sum > 11) includes only {12}; since these sets have no common outcomes, they are mutually exclusive.

The multiples of 5 between 1 and 110 are 5, 10, ..., 110. This is an arithmetic sequence: 110 = 5 + (n-1)5, so 110 = 5n, n = 22.

The spinner has 5 outcomes and the coin has 2 outcomes. Since these are independent events, the total number of combined outcomes is 5 * 2 = 10.

The jar contains 12 white, 10 grey, and 20 black marbles. The number of favourable outcomes for picking a black marble is simply the number of black marbles, which is 20.

Total ways = 52C4 = 270725. Ways to pick one from each suit = 13C1 * 13C1 * 13C1 * 13C1 = 13^4 = 28561. Probability = 28561 / 270725 = 2197 / 20825.

The sample space for n coins tossed is the set of all possible outcomes, which has 2^n elements. The sum of binomial coefficients nC0 + nC1 + ... + nCn equals 2^n, which represents the total number of subsets or outcomes.

The sum of 4 dice is 23. The only combinations are (6,6,6,5) in 4 permutations. Total outcomes are 6^4 = 1296. Probability = 4/1296 = 1/324.

P(Head) = 1/2. P(4) = 1/6. P(Head or 4) = P(Head) + P(4) - P(Head and 4). Since they are independent, P(Head and 4) = 1/2 * 1/6 = 1/12. P = 1/2 + 1/6 - 1/12 = 6/12 + 2/12 - 1/12 = 7/12.

Probability of not drawing a spade in one draw is 39/52 = 3/4. For three draws with replacement, the probability of drawing no spades is (3/4)^3 = 27/64. The probability of at least one spade is 1 - 27/64 = 37/64.

Total marbles = 14+11+5 = 30. Yellow marbles = 11. Not yellow = 30-11 = 19. Probability = 19/30 = 0.6333.

Total cards = 52. Ram takes one card (Queen of Spades). Remaining cards = 51. Total spades remaining = 13 - 1 = 12. Probability that Shyam draws a spade = 12 / 51 = 4 / 17.