Probability

Total marbles = 15. Probability of picking no blue marbles in 4 draws is (11/15) * (10/14) * (9/13) * (8/12) = 22/91. The probability of at least one blue is 1 - 22/91 = 69/91.

The probability of an event is the number of favorable outcomes divided by the total number of trials. Here, the probability of getting a head is 120/200, which simplifies to 12/20 or 3/5.

Total ways to select 4 balls from 10 is 10C4 = 210. Ways to select 4 balls with no white balls (all black) is 6C4 = 15. Ways with at least one white ball = 210 - 15 = 195. Quantity A = 195, Quantity B = 195. They are equal.

There are 4 kings in a deck of 52 cards. Probability = 4/52 = 1/13.

Total outcomes = 6^3 = 216. Outcomes with exactly two dice showing the same number: Choose the number that repeats (6 ways), choose the two dice that repeat (3 ways), and choose the number for the third die (5 ways). Total = 6 * 3 * 5 = 90. Probability = 90/216 = 5/12.

Total marbles = 3+4+6+2 = 15. Non-blue marbles = 12. Probability of picking 4 non-blue = C(12,4) / C(15,4) = 495 / 1365 = 33/91.

Total marbles = 3+4+6+2 = 15. Ways to choose 3: 15C3 = (15*14*13)/(3*2*1) = 455. Ways to choose 1 yellow (2C1) and 2 red (4C2): 2 * 6 = 12. Probability = 12/455.

Total balls = 9. Ways to pick 2 balls = 9C2 = 36. Ways to pick different colors (1 white, 1 red) = 5C1 * 4C1 = 20. Probability = 20/36 = 5/9.

The number of ways to choose 2 red balls from 5 is C(5,2) = 10. The number of ways to choose 2 green balls from 4 is C(4,2) = 6. Total ways to choose 4 balls from 9 is C(9,4) = 126. Probability = (10 * 6) / 126 = 60 / 126 = 10 / 21.

The probability is C(15,2) × C(10,1) divided by C(25,3). This equals 1050/2300 = 21/46.

Total balls = 6+11+5 = 22. P(Red first) = 6/22 = 3/11. P(Yellow second | Red first) = 11/21. Total probability = (3/11) * (11/21) = 3/21 = 1/7.

The probability of rolling a 2 is 1/6. The probability of rolling a 5 is 1/6. Since these are mutually exclusive, the probability of rolling either is 1/6 + 1/6 = 2/6.

Probability of choosing a bag is 1/2. In bag 1 (1, 2, 3, 4), P(4) = 1/4. In bag 2 (2, 4, 6, 7, 8, 9), P(4) = 1/6. Total probability = (1/2 * 1/4) + (1/2 * 1/6) = 1/8 + 1/12 = 3/24 + 2/24 = 5/24.