Not when both have positive probability. In the worked example below, learning that A occurred rules out B, even though B was possible beforehand. That is different from saying that the information about A leaves B unchanged. The positive-probability condition matters; a zero-probability event needs a separate check.
A finite example
Hypothetical model: one token is selected uniformly from the six tokens numbered 1 through 6. Define event A as selecting 1 or 2, and event B as selecting 3 or 4. The six elementary outcomes are equally likely, so:
| Quantity | Calculation | Value |
|---|---|---|
| P(A) | 2 ÷ 6 | 1/3 |
| P(B) | 2 ÷ 6 | 1/3 |
| P(A and B) | 0 ÷ 6 | 0 |
| P(A)P(B) | (1/3)(1/3) | 1/9 |
A and B are mutually exclusive because no single token belongs to both sets. However, independence requires the intersection probability to equal the product of the two event probabilities. Here, 0 is not equal to 1/9, so these events are not independent.
What the information changes
Before any selection, event B has probability 1/3. If you are told that A occurred, the possible result is either 1 or 2. Neither result is in B, so the conditional probability of B after learning A is 0. The information has changed the probability from 1/3 to 0; that is dependence in a direct, calculable form.
Do not replace “the same draw” with “two separate draws” midway through this question. That would introduce another experiment. Here A and B describe the single selected token, so a result of 1 cannot also be 3 or 4.
The zero-probability limit
Suppose P(A)=0. Then P(A and B)=0, while P(A)P(B)=0×P(B)=0. The multiplication condition can therefore hold, although A cannot provide ordinary information about B because A has no probability in the model. Thus, with positive probabilities for both events, mutually exclusive events cannot be independent; without that positivity condition, the blanket statement needs this qualification.
Reference: OpenStax explanation of independent and mutually exclusive events.
For a different question about interpreting past observations, see our history-screen reading note. The hypothetical counts above are not observed results.