Final answer:
To find the probability that the sum of numbers on two rolled dice is even, calculate that both dice show even numbers, and that both show odd numbers, and then add these probabilities. The result is that the probability for an even sum is 1/2.
Step-by-step explanation:
To calculate the probability that the sum of the numbers on two rolled dice is even, first understand that for any dice roll, there are a total of 36 possible outcomes (6 faces on the first die × 6 faces on the second die).
An even sum can occur in different ways: both dice show an even number, or both dice show an odd number. The probability of rolling an even number on one die is 3/6 (for 2, 4, or 6), and the probability of rolling an odd number is also 3/6 (for 1, 3, or 5). The product rule of probability is used to determine the likelihood of two independent events occurring together.
Using the product rule:
- The probability of both dice showing even is (3/6) × (3/6) = 9/36.
- The probability of both dice showing odd is also (3/6) × (3/6) = 9/36.
To find the total probability that the sum is even, you add the probabilities of these two cases together because they are mutually exclusive, thus applying the sum rule.
Total probability for an even sum = 9/36 + 9/36 = 18/36 = 1/2.
Therefore, the probability that the sum of the numbers on two rolled dice is even is 1/2.