On this page
Compare the probability conditions
| Relationship | Probability condition |
|---|---|
| Mutually exclusive | The intersection is empty, so its probability is zero |
| Independent | P(A ∩ B) = P(A)P(B) |
State the positive-probability qualification
If P(A) and P(B) are both positive, their product is positive. Mutually exclusive events instead have joint probability zero, so they cannot satisfy independence. This conclusion needs the positive-probability condition; “never independent” without qualification is too broad.
Check a clear zero-probability example
Let A be the empty event and B be heads on a fair coin. P(A) = 0 and P(B) = 0.5. Their intersection is empty, and 0 = 0×0.5, so they satisfy the joint-probability independence condition as well as mutual exclusivity. This example uses an empty event; it does not claim that every probability-zero event in every model is impossible.
Use conditional statements only when their denominators exist
The usual identity P(A|B) = P(A∩B)/P(B) requires P(B) > 0. The joint-probability definition of independence avoids division by zero. Do not substitute into the conditional formula with a zero denominator and then call the resulting expression a valid probability.