Problem 7 - A First Course in Probability

Question

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

Step-by-Step Solution

Answer

The short answer based on the given scenarios is as follows: (a) The possible maximum values of the two rolls are \( \{1, 2, 3, 4, 5, 6\} \). (b) The possible minimum values of the two rolls are \( \{1, 2, 3, 4, 5, 6\} \). (c) The possible sums of the two rolls are \( \{ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \). (d) The possible values for the first roll minus the second roll are \( \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \).

Step 1: (a) Maximum value

To determine the maximum value of the two rolls, we will consider the possible outcomes when we roll the die twice. There are 6 x 6 = 36 possible outcomes, where the maximum value can vary from 1 to 6. The possible maximum values are [1, 2, 3, 4, 5, 6], which is when we have rolls like (1,1), (1,2) or (2,1), (2, 3) or (3,2), (3,4) or (4,3), (4,5) or (5,4), (5,6) or (6,5), and (6,6), respectively.

Step 2: (b) Minimum value

Similarly, to find the minimum value of the two rolls, we will consider the possible outcomes when rolling a die twice. The possible minimum values are also [1, 2, 3, 4, 5, 6], which we can get from rolls like (1,1), (1,2) or (2,1), (1,3) or (3,1), (1,4) or (4,1), (1,5) or (5,1), and (1,6) or (6,1), respectively.

Step 3: (c) Sum of the rolls

For the sum of the two rolls, we can have values ranging from 2 to 12. The possible sums are [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], which are obtained from rolls like (1,1), (1,2) or (2,1), (1,3) or (3,1), (1,4) or (4,1), (1,5) or (5,1), (1,6) or (6,1), (2,6) or (6,2), (3,6) or (6,3), (4,6) or (6,4), (5,6) or (6,5), and (6,6), respectively.

Step 4: (d) Difference between the rolls

Lastly, for the difference between the first roll and the second roll, the values can range from -5 to 5. The possible differences are [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5], which can be obtained from rolls like (1,6), (2,6) or (1,5), (3,6) or (2,5) or (1,4), (4,6) or (3,5) or (2,4) or (1,3), (5,6) or (4,5) or (3,4) or (2,3) or (1,2), (6,6) or (5,5) or (4,4) or (3,3) or (2,2) or (1,1), (6,5) or (5,4) or (4,3) or (3,2) or (2,1), (6,4) or (5,3) or (4,2) or (3,1), (6,3) or (5,2) or (4,1), (6,2) or (5,1), and (6,1), respectively.