# Events One of Which equals Union

## Theorem

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.

Let $A$ and $B$ be such that:

- $A \cup B = A$

Then whenever $B$ occurs, it is always the case that $A$ occurs as well.

## Proof

From Union with Superset is Superset:

- $A \cup B = A \iff B \subseteq A$

Let $B$ occur.

Let $\omega$ be the outcome of $\EE$.

Let $\omega \in B$.

That is, by definition of occurrence of event, $B$ occurs.

Then by definition of subset:

- $\omega \in A$

Thus by definition of occurrence of event, $A$ occurs.

Hence the result.

$\blacksquare$

## Examples

### Target of Concentric Circles

Let $T$ be a target which consists of $10$ concentric circles $C_1$ to $C_{10}$ whose radii are respectively $r_k$ for $k = 1, 2, \ldots, 10$.

Let $r_k < r_{k + 1}$ for all $k = 1, 2, \ldots, 9$.

That is, let $C_1$ be the innermost and $C_{10}$ be the outermost.

Let $A_k$ denote the event of hitting $T$ inside the circle of radius $r_k$.

Let $B$ denote the event:

- $B = \displaystyle \bigcup_{k \mathop = 1}^6 A_k$

Then $B$ is the event of hitting $T$ inside circle $C_6$.

## Sources

- 1968: A.A. Sveshnikov:
*Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions*(translated by Richard A. Silverman) ... (previous) ... (next): $\text I$: Random Events: $1$. Relations among Random Events: Example $1.1$