First, let's consider all of the pairs that could be chosen from R and Q. There are sixteen possibilities:

1 and 2; 1 and 6; 1 and 8; 1 and 24; 3 and 2; 3 and 6; 3 and 8; 3 and 24; 4 and 2; 4 and 6; 4 and 8; 4 and 24; 12 and 2; 12 and 6; 12 and 8; 12 and 24.

Now, we need to find the product of each of these pairs, and then determine whether or not they belong to Q. The products of all of the possible pairs would be as follows:

1(2) = 2, which belongs to Q

1(6) = 6, which belongs to Q

1(8) = 8, which belongs to Q

1(24) = 24, which belongs to Q

3(2) = 6, which belongs to Q

3(6) = 18, which doesn't belong to Q

3(8) = 24, which belongs to Q

3(24) = 72, which doesn't belong

4(2) = 8, which belongs to Q

4(6) = 24, which belongs to Q

4(8) = 32, which doesn't belong

4(24) = 96, which doesn't belong

12(2) = 24, which belongs to Q

12(6) = 72, which doesn't belong

12(8) = 96, which doesn't belong

12(24) = 288, which doesn't belong

Thus, of our sixteen possible combinations, there are 9 which, when multiplied, give a number that belongs to Q. Because probability is the number of successful events out of the total number of events, the probability is 9/16.

The answer is 9/16.

Only John's statement must be true. Suzy could have freckles and not be Tim's sibling and if Jake doesn't have freckles then he is not Tim's sibling.

The probability that a blue marble is chosen first is 5/10 because at the outset there are 5 blue marbles in a total of 10 marbles.  After one marble has been chosen, there are 9 total marbles remaining since there is no replacement.  The probability that a red marble is chosen in the second pick is 3/9 (because there are 3 red marbles out of a total of 9).  There are now 8 total marbles remaining and 2 green marbles. Thus, the probability of picking a green marble on the third pick is 2/8. 

Therefore, the probability of picking a blue-red-green marble outcome (in that order) is:

5/10 * 3/9 * 2/8 = 30/720 = 1/24

Thinking through our data, we know that we are looking for the following combination of events, where T is a theology book, and < > represents a draw from the shelf:

1. <T><T>

2. <T><Non-T>

3. <Non-T><T>

To save us quite a bit of trouble, let us note that only one event is excluded:

<Non-T><Non-T> 

The easiest way to solve this would be to solve for the probability of this one case and subtract that from 1. This will give us the "remaining probability" that applies to the three cases that we want.

The <Non-T><Non-T> case would be calculated:

First draw: 25/40 = 5/8

Second draw: 24/39 = 8/13

Total probability: (5/8) * (8/13) = 5/13

The probability of our case is 1 – (5/13) = (13 – 5)/13 = 8/13.

You multiply each number of students in each class by how many times that class is entered into the drawing, then add up the totals. There will be a total of 1800 entrants and 800 will be seniors.

There are four possible numbers that can be chosen from S, and there are four that can be chosen from T. This means that there are 4 * 4, or 16, pairs of numbers that can be drawn from S and T. We need to find the sum of these pairs and determine if each sum is prime. Let's find the sum of the 16 pairs. Remember that a number is prime if it divisible only by itself and 1.

0 + 2 = 2, which is prime

0 + 3 = 3, prime

0 + 4 = 4, not prime

0 + 7 = 7, prime

1 + 2 = 3, prime

1 + 3 = 4, not

1 + 4 = 5, prime

1 + 7 = 8, not

5 + 2 = 7, prime

5 + 3 = 8, not

5 + 4 = 9, not

5 + 7 = 12, not

9 + 2=11, prime

9 + 3 = 12, not

9 + 4 = 13, prime

9 + 7 = 16, not

Of the sixteen pairs, 8 have a sum that equals a prime number. Thus, because each of these pairs has an equal chance of being drawn randomly, the probability that the sum will be prime is 8 out of 16, or 8/16 = 1/2.

The answer is 1/2.

We can find the individual probabilities of these three events occuring first. 

P(rolling a 6) = 1/6

P(head) = 1/2

P(spade) = 13/52 = 1/4

Now, to find the probability of rolling a 6 AND flipping a head AND drawing a spade, we must multiply the individual probabilities. So the answer is 1/6 * 1/2 * 1/4 = 1/48.

There are a total of 7 + 5 + 3 + 6 = 21 marbles. The probability of picking a yellow marble is 7/21 = 1/3. Then we put it back and choose a blue marble with probability 3/21 = 1/7. We do NOT put this blue marble back, but then we grab for a white. The probability of picking a white is now 6/20 = 3/10, because now we are choosing from 20 marbles instead of 21. So putting it together, the probability of choosing a yellow marble, replacing it and then choosing a blue and a white, is 1/3 * 1/7 * 3/10 = 1/70. 

There are 4 aces in the 52 card deck, so the probability of drawing an ace is 4/52. Then we put this ace back in the deck and draw again. The probability of drawing an ace is again 4/52. Similarly, on the third draw, the probability of getting an ace is 4/52. So the probability of drawing an ace on the 1st draw AND the 2nd draw AND the 3rd draw is 4/52 * 4/52 * 4/52.