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

## Source Work

1968: A.A. Sveshnikov: *Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions* (translated by Richard A. Silverman):

- $\text I$ Random Events
- $1$. Relations among Random Events
- Problem $3$

- $1$. Relations among Random Events

## Mistake

*A target consists of $10$ concentric circles of radius $r_k (k = 1, 2, 3, \ldots, 10)$. An event $A_k$ means hitting the interior of a circle of radius $r_k (k = 1, 2, \ldots, 10)$. What do the following events mean?**$\ds B = \bigcup_{k \mathop = 1}^6 A_k, \qquad C = \prod_{k \mathop = 5}^{10} A_k$?*

Note that in the above, $\ds C = \prod_{k \mathop = 5}^{10} A_k$ is the notation that A.A. Sveshnikov uses for what we on $\mathsf{Pr} \infty \mathsf{fWiki}$ would write $\ds C = \bigcap_{k \mathop = 5}^{10} A_k$.

## Correction

The question fails to state whether the circle radius $r_1$ or the circle radius $r_{10}$ is the innermost.

It is not obvious: it is feasible for the circles to have been numbered according to the score that the archer would achieve, in which case $r_{10}$ would be innermost.

However, that arrangement would be inconsistent with the answer given in the back of the book, and so it is apparent that:

- $r_k < r_{k + 1}$

for $k = 1, 2, \ldots, 9$.

## 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 $3$