Mutually Exclusive Events

A single roll of a die can't land on 3 and 5 at once. Those two outcomes live in completely separate spaces: one result excludes the other automatically. That's what mutually exclusive means — the events share no outcomes in common. Picture a Venn diagram with two circles for event A and event B. If the events are mutually exclusive, the circles don't touch. No overlap means nothing gets double-counted.

This is why the general addition rule, P(A or B) = P(A) + P(B) - P(A and B), collapses into something simpler. Since mutually exclusive events can never occur together, P(A and B) = 0, and the subtraction term disappears. You're left with P(A or B) = P(A) + P(B). So if P(A) = 0.3 and P(B) = 0.45, then P(A or B) = 0.75, no overlap to worry about.

The trap is confusing mutually exclusive with independent — they are not the same idea, and in fact they usually pull in opposite directions. Independence means knowing one event happened tells you nothing about the other. Mutually exclusive events do the opposite: if A happens, you know for certain B did not. That's a strong dependence, not an absence of one. This distinction matters going forward, because later rules for combining probabilities — the general addition rule and the multiplication rule for independent events — depend on correctly identifying which relationship, if any, connects two events.