two dice are rolled. their sum is recorded. what is the probability that their sum is between five and ten?

To find the probability that the sum of two rolled dice is between five and ten, we need to determine the number of ways to achieve the desired outcomes, and divide it by the total possible outcomes.

First, let's calculate the total number of possible outcomes when rolling two dice. Each die has six possible outcomes (numbers one through six), so the total number of outcomes for two dice is 6 multiplied by 6, which equals 36.

Now, let's find the number of favorable outcomes, i.e., the outcomes where the sum is between five and ten (inclusive).

To do this, we can list out all the possible combinations and count the outcomes that satisfy the condition:
- Possible combinations for a sum of five: (1, 4), (2, 3), (3, 2), (4, 1) (which amounts to four outcomes).
- Possible combinations for a sum of six: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) (five outcomes).
- Possible combinations for a sum of seven: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) (six outcomes).
- Possible combinations for a sum of eight: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) (five outcomes).
- Possible combinations for a sum of nine: (3, 6), (4, 5), (5, 4), (6, 3) (four outcomes).
- Possible combinations for a sum of ten: (4, 6), (5, 5), (6, 4) (three outcomes).

Summing up all these outcomes, we get a total of 4 + 5 + 6 + 5 + 4 + 3 = 27 favorable outcomes.

Finally, divide the number of favorable outcomes (27) by the total number of possible outcomes (36) to get the probability:
Probability = 27/36 = 3/4 = 0.75

Therefore, the probability that the sum of two rolled dice is between five and ten is 0.75 or 75%.