Yes—order matters if the question distinguishes which token was drawn first. It does not matter if the event only asks which two tokens appeared together. An event groups possible outcomes; see the OpenStax terminology reference.
Ordered outcomes: first draw and second draw are distinct
Hypothetical model: Four labelled tokens are available: A, B, C, and D. Two are drawn without replacement, and every legal ordered result is treated as equally likely. The first position records the first token; the second position records the next token.
| First token | Possible second tokens | Ordered outcomes |
|---|---|---|
| A | B, C, D | AB, AC, AD |
| B | A, C, D | BA, BC, BD |
| C | A, B, D | CA, CB, CD |
| D | A, B, C | DA, DB, DC |
The table contains 12 ordered outcomes. Only AB satisfies “A first, B second”, giving probability 1/12.
Unordered outcomes: the same two tokens form one pair
If the question is “Were A and B the two tokens drawn, in either order?”, AB and BA describe the same unordered pair: {A,B}. Grouping the ordered list gives six pairs:
| Unordered pair | Orders included | Count |
|---|---|---|
| {A,B} | AB, BA | 2 |
| {A,C} | AC, CA | 2 |
| {A,D} | AD, DA | 2 |
| {B,C}, {B,D}, {C,D} | BC/CB, BD/DB, CD/DC | 2 each |
Thus there are six unordered pairs. The event “the pair is {A,B}, in either order” contains two of the original 12 equally likely outcomes, so its probability is 2/12, which simplifies to 1/6.
Match the numerator to the denominator
The two answers are not contradictory. “A then B” is a more specific event than “A and B in either order.” If you use the ordered sample space, the denominator is 12 and the matching count for {A,B} is 2. If you use the grouped unordered description, the denominator is 6 and the matching count is 1. Both give 1/6.
Write the counting unit beside the fraction: ordered sequences or unordered pairs. “First A, then B” requires order. “A and B, either order” does not.
For a different question about interpreting past observations, see our history-screen reading note. The hypothetical counts above are not observed results.