Probability of a Sum of 7 with Two Dice: Why It Is 1/6

What is the probability of a sum of 7?

For two independent fair six-sided dice, the probability that their sum is 7 is 6/36 = 1/6. There are 36 equally likely ordered pairs and six favourable pairs: (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1).

Why there are 36 outcomes

Think of one die as red and the other as blue. Each of the six red outcomes can pair with six blue outcomes, giving 6 × 6 = 36. The colours are just labels; they let us distinguish which result belongs to which die. The same counting works when one die is thrown twice independently.

The possible sums are 2 through 12, but these eleven sums are not equally likely. Sum 2 has only (1,1), whereas sum 7 has six pairs. Dividing by 11 would incorrectly treat a rare sum and a frequent sum alike.

Compare sum 8 and a double

Sum 8 occurs through (2,6), (3,5), (4,4), (5,3) and (6,2). Its probability is 5/36. A double occurs through (1,1) to (6,6), giving 6/36 = 1/6. Having equal probabilities does not mean these are the same event.

For sum 7 or a double, the two events do not overlap, so add their counts: 12/36 = 1/3. For other combinations, check overlap before adding. For example, sum 8 and a double share (4,4).

Practice

1. Sum 5: four pairs, so 1/9.
2. Sum 12: only (6,6), so 1/36.
3. Sum 8 or a double: 5 + 6 − 1 = 10 pairs, so 5/18.

List the pairs systematically in increasing order of the first die. This reduces missed outcomes and double counting. If the dice are biased or dependent, the simple favourable-count/36 method is not justified.

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