Union of Events/Examples/Defective Devices
Jump to navigation
Jump to search
Examples of Unions of Events
Consider the experiment $\EE$ such that $3$ devices are checked as to whether they are operational.
Let $A$ be the event that at least $1$ of these $3$ devices is defective.
Let $B$ be the event that all $3$ devices are sound.
Then their union $A \cup B$ is a certainty.
Proof
Let $A$ occur.
Then it is not the case that all $3$ devices are sound.
Therefore $B$ does not occur.
Let $B$ occur.
Then it is not the case that at least $1$ device is defective.
Therefore $A$ does not occur.
Thus it is seen that:
- $B = \overline A$
that is, $B$ is the complementary event to $A$.
Thus:
- $A \cup B$
is the same thing as:
- $A \cup \overline A$
and the result follows from Union of Event with Complement is Certainty.
$\blacksquare$
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 $4 \ \text {(a)}$