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Detailed Solution

295. d: The correct answer was 5/6.
Let E be the event "the first die is a 5" and F be the event "the second die is a 2." E and F are independent events. Hence so are the complements of E and F. This means that the probability of the complement of E given the complement of F is just the probability of the complement of E, which is 1 - 1/6 = 5/6.

296. c: The correct answer was 64/1323.
Let x = the probability of rolling a 6. Then 2x is the probability of rolling a 5, 2*2x = 4x = the probability of rolling a 4, 8x = the probability of rolling a 3, 16x = the probability of rolling a 2, and 32x = the probability of rolling a 1. These are all the outcomes, so these probabilities must sum to 1: x + 2x + 4x + 8x + 16x + 32x = 1 which simplifies to 63x = 1 which means x = 1/63. So Pr(1) = 32/63, Pr(2) = 16/63, Pr(3) = 8/63, Pr(4) = 4/63, Pr(5) = 2/63, and Pr(6) = 1/63.
There are six ways to roll a sum of 7:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
By independence, their corresponding probabilities are
(32/63)*(1/63), (16/63)*(2/63), (8/63)*(4/63), (4/63)*(8/63), (2/63)*(16/63), and (1/63)*(32/63).
Each probability is 32/63^2. We add up these six probabilities to get the probability of rolling a sum of 7: 6*(32/63^2) = 64/1323.

297. a: The correct answer was 5/12.
There are 36 possible results from rolling two dice. The sum of the dice rolls will give you a number from 2 to 12. The prime numbers between them are 2, 3, 5, 7, and 11. The possible dice rolls to get those numbers are: 1-1, 1-2, 2-1, 1-4, 4-1, 2-3, 3-2, 1-6, 6-1, 2-5, 5-2, 4-3, 3-4, 6-5, and 5-6. Since there are 15 results that win the game, the odds of winning are 15/36, or 5/12.

298. c: The correct answer was 11 out of 850.
The probability of drawing one spade is 13/52 or 1/4. The odds of drawing another spade is 12/51. The odds of drawing a third spade is 11/50. If you multiply (1/4 * 12/51 * 11/50), you get 132/10200. This simplifies to 11/850.

299. a: The correct answer was 3750/7776.
There are 36 possible results from rolling two dice. Of those results, there are 6 that will give you a 7: 1-6, 6-1, 2-5, 5-2, 4-3, and 3-4. This means there is a 1/6 chance you will roll a 7. To figure the odds of making money, first I'll figure the odds of losing money. You have to lose 4 times in order to lose money. The odds of losing once are 5/6. If you do 5/6 to the 4th power, you get the odds of losing money, which are 4026/7776. This means the odds of making money are 3750/7776.

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