Probability

A die has outcomes {1, 2, 3, 4, 5, 6}. Composite numbers are {4, 6}. There are 2 composite numbers out of 6 total outcomes. Probability = 2/6 = 1/3.

Total balls = 5 + 3 + 2 = 10. Number of red balls = 5. Probability = 5/10 = 1/2.

The only outcome excluded from “at most two tails” is three tails. Its probability is (1/2)^3 = 1/8, so the required probability is 1 - 1/8 = 7/8.

Total outcomes = 6^3 = 216. Sum of 8: (1,1,6)x3, (1,2,5)x6, (1,3,4)x6, (2,2,4)x3, (2,3,3)x3. Total favorable = 3+6+6+3+3 = 21. Probability = 21/216 = 7/72.

The probability of getting exactly k successes in n trials is given by the binomial formula: C(n, k) * p^k * (1-p)^(n-k). Here, n=10, k=6, and p=0.5. Calculating this gives C(10, 6) * (0.5)^10 = 210 / 1024, which simplifies to 105/512.

The probability of getting at least one head in n tosses is 1 - (1/2)^n. We need 1 - (1/2)^n >= 0.99, which simplifies to (1/2)^n <= 0.01. Since (1/2)^6 = 1/64 = 0.0156 and (1/2)^7 = 1/128 = 0.0078, the minimum n is 7.

There are 13 diamond cards in a 52-card deck. Probability of diamond = 13/52 = 1/4. Probability of NOT diamond = 1 - 1/4 = 3/4.

A pack has 52 cards, 26 are black and 26 are red. The probability of drawing a card that is not black (i.e., red) is 26/52 = 1/2.

Total cards = 52. Hearts = 13. Kings = 4. Heart-Kings = 1. Union of Hearts and Kings = 13 + 4 - 1 = 16. Neither heart nor king = 52 - 16 = 36. Probability = 36 / 52 = 9 / 13.