Union of Events/Examples/Defective Devices

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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