Occurrence of Event/Examples/Electric Circuit 1

Examples of Occurrences of Events

Consider the electric circuit:

Let event $A$ be that switch $A$ is open.

Let event $B_n$ for $n = 1, 2, 3$ be that switch $B_n$ is open.

Let $C$ be the event that no current flows from $M$ to $N$.

Then:

$C = A \cup \paren {B_1 \cap B_2 \cap B_3}$

$\overline C = \overline A \cap \paren {\overline {B_1} \cup \overline {B_2} \cup \overline {B_3} }$

The circuit is open between $M$ and $N$ if and only if:

switch $A$ is open

or:

all of $B_1$, $B_2$ and $B_3$ are open.

The corresponding events are:

$A$

and

$B_1 \cap B_2 \cap B_3$

from which $C = A \cup \paren {B_1 \cap B_2 \cap B_3}$ follows.

Then:

$\ds \overline C$

$=$

$\ds \overline {A \cup \paren {B_1 \cap B_2 \cap B_3} }$

$\ds $

$=$

$\ds \overline A \cap \overline {\paren {B_1 \cap B_2 \cap B_3} }$

$\ds $

$=$

$\ds \overline A \cap \paren {\overline {B_1} \cup \overline {B_2} \cup \overline {B_3} }$

$\blacksquare$