Question

An electrical system has open-closed switches $S_1,S_2$ and $S_3$ as shown. The switches operate independently of one another and the current will flow from A to B either if $S_1$ is closed or if both $S_2$ and $S_3$ are closed. If $P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=1/2,$ find the probability that the cricult will work.

Answer 1

$P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=\frac{1}{2}$

Then circuit will work when $S_1$ is closed or $S_2$ and $S_3$ both are closed.

Hence Total Probability that circuit will work$\frac{1}{2}+(\frac{1}{2}\times \frac{1}{2})=\frac{1}{2}+\frac{1}{4}=\frac{3}{4}$

Probability that circuit will work$=\frac{3}{4}$