Probability

To find the probability that the product of the rolls is 3 or 12, we can first list all the possible outcomes of rolling the die twice and calculate the product of each outcome:

1 × 1 = 1
1 × 2 = 2
1 × 3 = 3
1 × 4 = 4
1 × 5 = 5
1 × 6 = 6
2 × 1 = 2
2 × 2 = 4
2 × 3 = 6
2 × 4 = 8
2 × 5 = 10
2 × 6 = 12
3 × 1 = 3
3 × 2 = 6
3 × 3 = 9
3 × 4 = 12
3 × 5 = 15
3 × 6 = 18
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
4 × 6 = 24
5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 4 = 20
5 × 5 = 25
5 × 6 = 30
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
6 × 5 = 30
6 × 6 = 36

We can see that the only outcomes that have a product of 3 or 12 are:

1 × 3 = 3
3 × 1 = 3
2 × 6 = 12
6 × 2 = 12

Therefore, out of the 36 possible outcomes, there are only 4 outcomes that have a product of 3 or 12. Therefore, the probability of rolling a product of 3 or 12 is:

P(product is 3 or 12) = 4/36 = 1/9

Therefore, the probability that the product of the rolls is 3 or 12 is 1/9 or approximately 0.111.