Why does a probability calculation need a clearly defined set of outcomes?

You can list the totals to describe the result, but you can’t assume those 11 totals are equally likely. First decide what one outcome means in your model. For two consecutive rolls, one natural choice is an ordered pair: first-roll result, then second-roll result. For example, (2, 5) and (5, 2) are different outcomes because they record different orders. A sample space is the set of possible outcomes; its outcomes should be mutually exclusive and cover every possible result of the specified experiment. [S1]

Assume the die is fair and the rolls are independent. The ordered-pair model has 36 equally likely pairs. Six pairs add to 7, so the probability is 6/36, or 1/6.

If you instead treat totals 2 through 12 as 11 equally likely outcomes, you get 1/11. That is not the probability of the sum from two rolls: a total of 7 can arise from several pairs, while a total of 2 has just one. Listing totals is fine; treating them as equally likely is the mistake.

Assume two rolls of the same fair six-sided die, with independent results. The target event is “the sum is 7.”

1. Count ordered pairs: An outcome is (first roll, second roll), such as (1, 1) or (1, 2). There are 36 equally likely pairs. The pairs that total 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Result: 6/36 = 1/6.

1. Count totals, keeping their probabilities: The possible totals are 2 through 12, but they are not equally likely. Total 2 comes from one pair; total 3 from two pairs; total 7 from six pairs. So the probability of 7 is still 6/36 = 1/6.

Both descriptions give the same answer when probabilities are handled correctly. The second groups pairs by their total; it does not make the total categories equally likely. Equal-likelihood counting applies only when the outcomes being counted really are equally likely. [S1]

Counting 1 out of 11 would be right for a different experiment: one where a total from 2 through 12 is selected uniformly at random. That is not the experiment described above.

You can use totals directly if you know their probabilities. In this example, the probability of a total is the number of ordered pairs producing it divided by 36: total 7 has probability 6/36, while total 2 has probability 1/36. If your target includes several totals, add their probabilities.

The die’s fairness and the independence of the rolls are assumptions here, not conclusions from the outcome definitions. If those assumptions do not fit your experiment, the probabilities may differ.

Write down the experiment, what one outcome means, and the event you want to count. Check that every possible result is represented exactly once. Then check whether the outcomes you plan to count are equally likely; if not, use their probabilities rather than a simple favorable-over-total count. [S1]

For this example: “Experiment: roll a fair six-sided die twice independently. Outcome: ordered pair (first roll, second roll). Event: the numbers sum to 7. Count: 6 favorable pairs out of 36 equally likely pairs. Probability: 1/6.” If you choose totals instead, remember that the totals are grouped outcomes and are not equally likely.