Events One of Which equals Union/Examples/Target of Concentric Circles

Examples of Use of Events One of Which equals Union

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 $C_6$.

Proof

By the geometry of the situation:

$C_1 \subseteq C_2 \subseteq \cdots \subseteq C_6$

By Events One of Which equals Union:

\(\ds A_1 \cup A_6 = A_6\)

\(\iff\)

\(\ds A_1 \subseteq A_6\)

\(\ds A_2 \cup A_6 = A_6\)

\(\iff\)

\(\ds A_2 \subseteq A_6\)

\(\ds \)

\(\vdots\)

\(\ds \)

\(\ds A_5 \cup A_6 = A_6\)

\(\iff\)

\(\ds A_5 \subseteq A_6\)

The result follows.

$\blacksquare$

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: Problem $3$

Note that the question does not state whether $r_k < r_{k + 1}$ for all $k = 1, 2, \ldots, 9$ or $r_k > r_{k + 1}$ for all $k = 1, 2, \ldots, 9$. The interpretation made here is the one which provides the correct answer.