Event a: roll greater than a 5 Event b: roll something divisible by 2 or 4 or both

Question

An ordinary (fair) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.

Event a: roll greater than a 5

Event b: roll something divisible by 2 or 4 or both

Answer 1

To address your inquiry, we can break it down into the specific events you mentioned:

- Event A: The sum of the two rolls exceeds 5.

- Event B: The sum of the two rolls is divisible by 2 or 4 (or both).

First, let's consider the possible outcomes when rolling a die twice. The smallest sum you can get is 2 (1+1), and the largest is 12 (6+6). There are 36 possible combinations (6 outcomes from the first roll × 6 results from the second roll).

Event A Analysis:

For Event A, where the sum exceeds 5, the possible sums are 6 through 12. We'll calculate the number of combinations resulting in each sum.

Event B Analysis:

For Event B, the sum needs to be divisible by 2 or 4. This includes even numbers and multiples of 4. Since every multiple of 4 is also even, we must ensure the sum is an even number. The even sums in our range (2 to 12) are 2, 4, 6, 8, 10, and 12. We'll calculate the number of combinations resulting in each sum.

Let's proceed with calculating the number of combinations that satisfy each event.

The calculations reveal the following:

- For Event A (sum more significant than 5), 26 combinations satisfy this condition. These are the sums of 6 through 12, with varying counts for each sum based on the possible dice roll combinations.

- For Event B (sum divisible by 2), 18 combinations meet this criterion. These include the even sums of 2, 4, 6, 8, 10, and 12, with different counts for each sum based on the roll combinations.

The detailed breakdown of possible sums from 2 to 12 and their corresponding number of combinations is as follows:

- Sum of 2: 1 combination (1+1)

- Sum of 3: 2 combinations (1+2, 2+1)

- Sum of 4: 3 combinations (1+3, 2+2, 3+1)

- Sum of 5: 4 combinations (1+4, 2+3, 3+2, 4+1)

- Sum of 6: 5 combinations (1+5, 2+4, 3+3, 4+2, 5+1)

- Sum of 7: 6 combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1)

- Sum of 8: 5 combinations (2+6, 3+5, 4+4, 5+3, 6+2)

- Sum of 9: 4 combinations (3+6, 4+5, 5+4, 6+3)

- Sum of 10: 3 combinations (4+6, 5+5, 6+4)

- Sum of 11: 2 combinations (5+6, 6+5)

- Sum of 12: 1 combination (6+6)

These outcomes provide a comprehensive understanding of the possibilities for each event within the context of rolling a fair die twice.