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I need to determine the probability that the current could go from left to right given that there is a probability of

*p*for each independent switch named a,b,c,d and e.

I have tried to solve this but I have no idea which part I am getting off the right track. Here is what I am trying to do.

Case I: Assume Switch E is closed so I have a probability of

P((AB) or (CD))=P(AB)+P(CD)-P(AB and CD)=2p^2-p^4

Case II: Assume Switch E is opened so I have a probability of

P((A or C) and (B or D))=(P(A)+P(C)-P(AC))(P(B)+P(D)-P(BD))=(2p-p^2)^2

so overall probability is (1-p)(2p^2-p^4)+p(2p-p^2)^2 which could be simplified to 2p^5-5p^4+2p^3+2p^2

however some friends of mine told be that it should be p^5-5p^4+2p^3+2p^2. I hope to know what's wrong in my calculation. Thanksss!