Probability

(i) Probability of 3 not coming up in one throw is 5/6. For two throws, (5/6)^2 = 25/36. (ii) Probability of 6 coming up at least once = 1 - P(no 6) = 1 - (5/6)^2 = 1 - 25/36 = 11/36.

Total outcomes for 3 coins = 2^3 = 8. Outcomes with exactly two heads: HHT, HTH, THH (3 outcomes). Probability = 3/8.

A die has 6 faces (1, 2, 3, 4, 5, 6). Numbers greater than 4 are 5 and 6. Probability = 2/6 = 1/3.

Each flip has 2 outcomes. For 4 flips, the total number of sample points is 2^4 = 16.

Total ways to pick 2 cards = 52C2 = (52*51)/2 = 1326. Ways to pick 2 diamonds = 13C2 = (13*12)/2 = 78. Probability = 78/1326 = 1/17.

The die has 6 faces in total. The number 2 appears on 3 of those faces. Therefore, the probability is 3/6, which simplifies to 1/2.

There are 16 coins total: 2 are double-headed (probability of head = 1) and 14 are fair (probability of head = 0.5). Total probability = (2/16)*1 + (14/16)*0.5 = 2/16 + 7/16 = 9/16.

To keep composition unaltered, either a red ball is moved and then a red ball is returned, or a black ball is moved and then a black ball is returned. P(R1, R2) = (3/5)(3/6) = 9/30. P(B1, B2) = (2/5)(4/6) = 8/30. Total probability = 9/30 + 8/30 = 17/30.

Outcomes: HH (1/4), HT/TH (2/4), TT (1/4). Expected value = (1/4 * 2) + (2/4 * 1) + (1/4 * -5) = 0.5 + 0.5 - 1.25 = -0.25 = -1/4.

Total cards = 52. Spades = 13. Aces = 4. Ace of spades is counted in both, so total favorable outcomes = 13 + 4 - 1 = 16. Probability = 16/52 = 4/13.

A wins on 1st toss (1/2), 3rd (1/2 * 1/2 * 1/2 = 1/8), 5th (1/2^5 = 1/32), etc. This is a geometric series with a = 1/2 and r = 1/4. Sum = a / (1 - r) = (1/2) / (1 - 1/4) = (1/2) / (3/4) = 2/3.

When tossing three coins, there are 2^3 = 8 total outcomes. The outcomes with exactly two heads are (H,H,T), (H,T,H), and (T,H,H), which is 3 favorable outcomes. Thus, the probability is 3/8.