Probability

Squares between 1 and 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. There are 10 such numbers. Probability = 10 / 100 = 1/10.

There are 216 equally likely outcomes when three dice are rolled. Favorable outcomes satisfy red + blue = green, and the possible sums from 2 through 6 contribute 1 + 2 + 3 + 4 + 5 = 15 outcomes, so P = 15/216 = 5/72 and a + b = 77.

Total probability must be 1. P(Red) + P(Blue) + P(White) = 1. 2/13 + 4/13 + P(White) = 1. 6/13 + P(White) = 1. P(White) = 7/13.

Even numbers on a die are 2, 4, 6. There are 3 even numbers out of 6 total outcomes. Probability = 3/6 = 1/2.

Total outcomes = 36. Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 outcomes. Sum of 10: (4,6), (5,5), (6,4) = 3 outcomes. Total favorable = 8. Probability = 8/36 = 2/9.

Total balls = 6 + 5 = 11. Black balls = 6. Probability = 6/11.

A standard deck has 52 cards, with 4 suits. Each suit has 13 cards. The probability of drawing a diamond is 13/52 = 1/4.

There are 4 queens and 4 jacks. Total = 8 favorable cards. Probability = 8 / 52 = 2 / 13.

There are 10 odd numbers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19) out of 20. Probability of first being odd = 10/20 = 1/2. Probability of second being odd = 9/19. Total probability = (1/2) * (9/19) = 9/38.

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

There are 4 Queens in a standard deck of 52 cards. The probability is 4/52, which simplifies to 1/13.

Binomial probability: P(X=4) = 5C4 * (1/2)^4 * (1/2)^1 = 5 * (1/16) * (1/2) = 5/32.

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

The probability of drawing a green marble is 2/3, so the probability of drawing a blue marble is 1 - 2/3 = 1/3. With 24 total marbles, the number of blue marbles is 24 * (1/3) = 8.