Question

An electrical system has open-closed switches $S_1,S_2,\text{and}{S}_3$ as shown in fig

The switch operate independently of one another and the current will flow $A$ to $B$ either if $S_1$ is closed or of both $S_2\text{and}{S}_3$ are closed. If $P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=\frac{1}{2}$, then the probability that the circuit will work is:

• A. $\frac{5}{8}$
• B. $\frac{5}{4}$
• C. $\frac{5}{16}$
• D. $\frac{7}{8}$

Answer 1

The correct option is A $\frac{5}{8}$

$P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=\frac{1}{2}$
Let $E$ be the event that the current will flow. Then,
$P\left(E\right)=P\left(\right(S_2\cap S_3\left)\text{or}{S}_1\right)$
$=P(S_2\cap S_3)+P\left(S_1\right)-P(S_1\cap S_2\cap S_3)$
$=\frac{1}{4}+\frac{1}{2}-\frac{1}{8}=\frac{5}{8}$