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To understand the concept of even and odd sums, we need to first grasp what makes a number even or odd. An even number is divisible by 2 without any remainder (e.g., 2, 4, 6), while an odd number will have a remainder of 1 when divided by 2 (e.g., 1, 3, 5). When adding two numbers:

• Two even numbers always result in an even sum. For example, 2 + 4 = 6.
• Two odd numbers also result in an even sum. For example, 1 + 3 = 4.
• An even and an odd number together result in an odd sum. For example, 2 + 3 = 5.

So, for the sum of numbers on two dice to be even, both rolled numbers must be either even or both must be odd.

Probability is a measure of how likely an event is to occur. When rolling two dice, each die has 6 faces, resulting in a total of 6 × 6 = 36 possible outcomes. Each combination is equally likely. Knowing the total number of outcomes helps us calculate the likelihood of specific events, such as rolling certain sums.

When examining the probability of obtaining an even sum with two dice, we consider the individual probabilities of rolling each face. Each die can show a number between 1 and 6. Since these outcomes are independent of one another, we multiply their probabilities when looking at combined events.

For understanding how probability works in simpler terms, think of the steps as follows:

• Count total possible outcomes (rolling two dice gives 36).
• Identify and count favorable outcomes (18 favorable pairs where the sum is even).
• Calculate the probability by dividing favorable outcomes by total outcomes: \(\frac{18}{36} = \frac{1}{2} \).

In probability, favorable outcomes are those that satisfy the condition of the event we are interested in. For the problem at hand, the event is getting an even sum from rolling two dice. Favorable outcomes include:

• Pairs where both numbers are even: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), and (6,6) - 9 combinations.
• Pairs where both numbers are odd: (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), and (5,5) - 9 combinations.

Combining these, we find 18 favorable outcomes. Since these are the combinations leading to an even sum, the probability of rolling an even sum with two dice is determined by dividing these favorable outcomes by the total number of possible outcomes, which is 36:

This results in a probability of \(\frac{18}{36} = \frac{1}{2} \), meaning there is a 50% chance of rolling an even sum with two dice.