A fair die is rolled twice with the two rolls being independent of each other. Let M be the maximum of the two rolls and D be the value of the first roll minus the value of the second roll. Are M and D independent?

To determine if M and D are independent, we need to check whether the probability distribution of one variable changes when the value of the other variable is known.

Let's consider the possible outcomes of rolling a fair die twice:

Outcomes: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), ..., (6,6)

Let's calculate the probability distribution for M, the maximum of the two rolls:

P(M = 1) = 1/36 (only one outcome: (1,1))
P(M = 2) = 2/36 (two outcomes: (1,2) and (2,1))
P(M = 3) = 3/36 (three outcomes: (1,3), (2,3), and (3,3))
P(M = 4) = 4/36 (four outcomes: (1,4), (2,4), (3,4), and (4,4))
P(M = 5) = 5/36 (five outcomes: (1,5), (2,5), (3,5), (4,5), and (5,5))
P(M = 6) = 6/36 (six outcomes: (1,6), (2,6), (3,6), (4,6), (5,6), and (6,6))

Now, let's calculate the probability distribution for D, the difference between the first roll and the second roll:

P(D = -5) = 1/36 (only one outcome: (1,6))
P(D = -4) = 1/36 (only one outcome: (2,6))
P(D = -3) = 1/36 (only one outcome: (3,6))
P(D = -2) = 1/36 (only one outcome: (4,6))
P(D = -1) = 1/6 (six outcomes: (1,2), (2,1), (2,3), (3,2), (3,4), and (4,3))
P(D = 0) = 1/6 (six outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6))
P(D = 1) = 1/6 (six outcomes: (1,2), (2,1), (3,2), (4,3), (5,4), and (6,5))
P(D = 2) = 1/36 (only one outcome: (1,3))
P(D = 3) = 1/36 (only one outcome: (1,4))
P(D = 4) = 1/36 (only one outcome: (1,5))
P(D = 5) = 1/36 (only one outcome: (1,6))

If M and D are independent, then the joint probability distribution should be equal to the product of the individual probability distributions.

Let's compare the probabilities for the joint distribution, P(M = m, D = d), with the product of the corresponding individual probabilities, P(M = m) * P(D = d):

P(M = 1, D = -5) = 1/36
P(M = 1) * P(D = -5) = (1/36) * (1/36) = 1/1296

These probabilities do not match, which means that M and D are not independent.

In conclusion, the maximum of the two rolls (M) and the difference between the first roll and the second roll (D) are not independent.

To determine if M and D are independent, we need to check if the probability of one event occurring is affected by the occurrence of the other event.

First, let's consider the possible outcomes of rolling two fair dice. Each die has six sides numbered from 1 to 6. So, the sample space consists of 36 equally likely outcomes, where each outcome represents a pair of numbers (x, y) where x and y are the numbers rolled on the first and second dice, respectively.

Next, let's define the random variables M and D in terms of these outcomes:
- M represents the maximum of the two rolls, so M takes on the values 1, 2, 3, 4, 5, or 6.
- D represents the difference between the first roll and the second roll, so D can take on values from -5 to 5.

Now, we can calculate the joint probability distribution of M and D, and check if it factors into the product of the individual probabilities.

To do this, we'll create a table of all possible outcomes and calculate the probabilities for each combination of M and D.

Here's the table of outcomes, where each entry represents the count of how many times that combination occurs:

| | D=-5 | D=-4 | D=-3 | D=-2 | D=-1 | D=0 | D=1 | D=2 | D=3 | D=4 | D=5 |
|---|------|------|------|------|------|-----|-----|-----|-----|-----|-----|
| M=1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| M=2 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| M=3 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
| M=4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
| M=5 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
| M=6 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |

To calculate the probabilities, we will divide each count by the total number of outcomes, which is 36.

For example, P(M=1 and D=-4) = 1/36, since there is only one outcome (1, 5) that satisfies both M=1 and D=-4.

Now, let's check if the joint probability distribution factors into the product of the individual probabilities.

For example, consider P(M=1) = 6/36, P(D=-4) = 2/36, and P(M=1 and D=-4) = 1/36. If M and D are independent, we should have P(M=1 and D=-4) = P(M=1) * P(D=-4). In this case, 1/36 ≠ (6/36)*(2/36), so we can conclude that M and D are dependent.

By performing similar calculations for all combinations of M and D, we can observe that the joint probability distribution does not factor into the product of the individual probabilities for any combination. Hence, M and D are not independent.