Question

If two switches $S_1$ and $S_2$ have respectively $90$ and $80$ chances of working. Find the probabilities that each of the following circuits will work.

Answer 1

Consider the following events:
A=Switch $S_1$ works,
B=Switch $S_2$ works,
We have,
$P\left(A\right)=\frac{90}{100}=\frac{9}{10}$ and $P\left(B\right)=\frac{80}{100}=\frac{8}{10}$
(i) The circuit will work if the current flows in the circuit. This is possible only when both the switches work together. Therefore, Required probability
$=P(A\cap B)=P\left(A\right)P\left(B\right)$ [$\because$ $A$ and $B$ are independent events]
$=\frac{9}{10}\times \frac{8}{10}=\frac{72}{100}=\frac{18}{25}$
(ii) The circuit will work if the current flows in the circuit. This is possible only when at least one of the two switches $S_1,S_2$ works. Therefore,
Required Probability
$=P(A\cup B)=1-P\left(\overline{A}\right)P\left(\overline{B}\right)$ [$\because$ $A,B$ are independent events]
$=1-(1-\frac{9}{10})(1-\frac{8}{10})=1-\frac{1}{10}\times \frac{2}{10}=\frac{49}{50}$