An electrical system has open-closed switches `S_1, S_2a n dS_3` as shown in fig. The switches operate independently of one another and the current will flow from `AtoB` either if `S_1` is closed or if both `S_2a n dS_3` are closed. If `P(S_1)=P(S_2)=P(S_3)=1//2,` then find the probability that the circuit will work. fig

Explanation

` P(S_(1))=P(S_(2))=(S_(3))=1/2`
Let E be event that "the current will flow".
`P(E)=P((S_(2)nnS_(3))nnS_(1)))`
`=P(S_(2)nnS_(3))+P(S_(3))-P(S_(1)nnS_(2)nnS_(3))`
`P(S_(2))P(S_(3))+P(S_(3))-P(S_(1))P(S_(2))P(S_(3))`
`=(1)/(2).(1)/(2)+(1)/(2)-(1)/(2).(1)/(2).(1)/(2).=5/8`