Consider a hypothetical experiment with two independent trials. Each trial can produce A or B, and both results are equally likely. The wording “at least one A” and “exactly one A” sounds similar, but it selects different outcomes. First define the sample space: the complete list of possible ordered results.
List the four possible outcomes
Because the first and second trial are recorded in order, the outcomes are AA, AB, BA, and BB. Under the stated assumptions, each has probability 1/4. The order matters here: AB means A first and B second, while BA means B first and A second.
| Outcome | Number of A results | At least one A? | Exactly one A? |
|---|---|---|---|
| AA | 2 | Yes | No |
| AB | 1 | Yes | Yes |
| BA | 1 | Yes | Yes |
| BB | 0 | No | No |
Calculate “at least one A”
“At least one” means one or more. Therefore, AA, AB, and BA qualify. There are three qualifying outcomes out of four:
P(at least one A) = 3/4 = 75%.
The complement gives a shorter check. The opposite of “at least one A” is “no A”, which is only BB. Thus 1 − P(BB) = 1 − 1/4 = 3/4. The complement contains outcomes outside the event. Reference: OpenStax’s probability terminology reference.
Calculate “exactly one A”
“Exactly one” excludes both zero and two. Only AB and BA qualify, so:
P(exactly one A) = 2/4 = 1/2 = 50%.
Using the complement of BB here would be incorrect, because removing only BB still leaves AA, which has two A results. The practical reading rule is simple: “at least one” includes every outcome meeting the minimum, while “exactly one” permits one and only one. These calculations describe this hypothetical two-trial model; past results do not alter its stated assumptions.
For a different question about interpreting past observations, see our history-screen reading note. The hypothetical counts above are not observed results.