Probability

Numbers 1-12. Odd numbers: 1, 3, 5, 7, 9, 11 (6 total). Probability = 6/12 = 1/2.

The total number of balls in the bag is 20 + 20 = 40. Since there are 20 red balls, the probability of drawing a red ball is 20/40, which simplifies to 1/2.

Prime numbers on a die are 2, 3, 5. There are 3 prime numbers out of 6 total outcomes. Probability = 3/6 = 1/2.

The probability that any specific ticket is drawn at any specific position (1st, 2nd, ..., 10th) is 1/10, because all positions are equally likely for any ticket.

Total cards = 101 - 2 + 1 = 100. Perfect squares between 2 and 101 are 4, 9, 16, 25, 36, 49, 64, 81, 100 (9 total). Probability = 9/100.

The probability of getting a head in one toss is 1/2. For 5 independent tosses, the probability of 5 heads is (1/2)^5 = 1/32.

For each toss, the probability of getting a head is 1/2 and a tail is 1/2. Getting no tails means getting all heads. The probability of getting heads in 6 consecutive tosses is (1/2)^6 = 1/64.

Die outcomes: {1, 2, 3, 4, 5, 6}. Composite numbers less than 5: {4}. Probability = 1/6.

The probability of getting at least one tail is 1 minus the probability of getting all heads. With 6 coins, the probability of all heads is (1/2)^6 = 1/64. Thus, 1 - 1/64 = 63/64.

Total outcomes = 6^3 = 216. Favorable outcomes (x < y < z): Choose 3 distinct numbers from 6, which is 6C3 = 20. Probability = 20/216 = 5/54.

The total outcomes when tossing two dice are 6 * 6 = 36. The pairs that sum to 5 are (1,4), (2,3), (3,2), and (4,1). There are 4 such outcomes. The probability is 4/36 = 1/9.

Total outcomes for 4 dice = 6^4 = 1296. Outcomes with no 3s = 5^4 = 625. Outcomes with at least one 3 = 1296 - 625 = 671.

The probability of losing a toss is 1/2. The probability of losing 13 times out of 14 is C(14, 13) * (1/2)^13 * (1/2)^1 = 14 * (1/2)^14 = 7 * (1/2)^13.