Equal likelihood applies to the complete elementary outcomes, not automatically to the labels created by grouping them. In the hypothetical model below, each number is selected independently and uniformly from 0, 1, and 2. The pair is ordered: the first position records the first selector and the second position records the second selector.
Start with the nine equally likely pairs
There are three choices for the first selector and three for the second, so the sample space contains 3 × 3 = 9 ordered pairs. Each pair has probability 1/9. Listing them prevents the five possible sums from being mistaken for five equally likely outcomes.
| Sum | Pairs producing it | Count | Probability |
|---|---|---|---|
| 0 | (0,0) | 1 | 1/9 |
| 1 | (0,1), (1,0) | 2 | 2/9 |
| 2 | (0,2), (1,1), (2,0) | 3 | 3/9 |
| 3 or 4 | (1,2),(2,1); (2,2) | 2; 1 | 2/9; 1/9 |
Why the middle sum is more frequent
Sum 2 is produced in three distinct ways: (0,2), (1,1), and (2,0). Since these are separate elementary outcomes, their probabilities add: 1/9 + 1/9 + 1/9 = 3/9. Sum 0 has only (0,0), so its probability remains 1/9. Nothing has made an individual pair more likely; sum 2 simply contains a larger event, meaning a larger set of elementary outcomes. Events may contain several outcomes; see the OpenStax terminology reference.
What “five possible sums” does and does not mean
The possible sums are 0, 1, 2, 3, and 4, but possibility alone does not establish equal probability. Equal probability would require each sum to contain the same number of equally likely pairs, which is not true here: the counts are 1, 2, 3, 2, and 1. The distribution is therefore concentrated toward the centre and smaller at the edges.
The key limit is that this conclusion belongs to this explicitly hypothetical model: independent uniform selectors with ordered pairs. If the selectors were not uniform, were dependent, or if a different rule grouped outcomes, the calculation would need to be rebuilt from its own sample space.
For a different question about interpreting past observations, see our history-screen reading note. The hypothetical counts above are not observed results.