If you roll a pair of dice once, which is a more likely outcome? An odd sum or an even sum, or are they both equally likely?

When rolling two standard six-sided dice, a total of 36 distinct outcomes are possible. Each die has faces numbered one through six, offering an equal chance for any number to appear. To understand the likelihood of an odd versus an even sum, we must consider the nature of the numbers on the dice.

An even sum can be achieved in two ways: either by rolling two even numbers or by rolling two odd numbers. For example, rolling a 2 and a 4 results in an even sum of 6, while rolling a 3 and a 5 gives an even sum of 8. An odd sum, on the other hand, requires one odd number and one even number. A roll of 1 and 4 yields an odd sum of 5, and a 6 and 3 results in an odd sum of 9.

Each individual die presents three odd outcomes (1, 3, 5) and three even outcomes (2, 4, 6). Consequently, there are 3x3, or 9, ways to roll two odd numbers, and 3x3, or 9, ways to roll two even numbers. This combines for a total of 18 ways to achieve an even sum. Similarly, there are 3x3, or 9, ways to roll an odd number then an even number, and another 3x3, or 9, ways to roll an even number then an odd number, also totaling 18 ways for an odd sum.

Therefore, with 18 possible combinations resulting in an odd sum and 18 possible combinations resulting in an even sum, both outcomes are equally likely when a pair of dice is rolled once. This symmetrical distribution highlights a basic principle of probability, demonstrating how understanding the parity of numbers can reveal the fairness of a chance event, a concept explored in games and mathematics for centuries.