# Disjoint Independent Events means One is Void

## Theorem

Let $A$ and $B$ be events in a probability space.

Suppose $A$ and $B$ are:

Then either $\map \Pr A = 0$ or $\map \Pr B = 0$.

That is, if two events are disjoint and independent, at least one of them can't happen.

## Proof

For $A$ and $B$ to be independent:

- $\map \Pr {A \cap B} = \map \Pr A \, \map \Pr B$

For $A$ and $B$ to be disjoint:

- $\map \Pr {A \cap B} = 0$

So:

- $\map \Pr A \, \map \Pr B = 0$

Hence the result.

$\blacksquare$

## Comment

If this makes you scratch your head in bewilderment, consider what it means.

For $A$ and $B$ to be independent, then whether one event occurs has no bearing on whether the other one occurs or not.

But suppose we know that they are disjoint.

That is, when one of them happens, the other *can't* happen.

Then they *can't* be independent, because the occurrence of one has a direct relation to the happening (or in this case, non-happening) of the other.

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.7$: Independent Events: Exercise $21$