A system with four switches is shown below. The probability that any of the switches operate...

Switches 1 & 2 are in series.

So,

P(Switches 1 & 2 together working) = p₁ p₂

Switches 3 & 4 are in series.

So,

P(Switches 3 & 4 together working) = p₃ p₄

Switches 1&2 and 3 & 4 are in parallel.

So,

P(Switches 1 & 2 and 3 & 4 together working) = 1 - {(1-p₁ p₂) (1 - p₃ p₄)}

Switches 1&2 and 3&4 and Switch 5 are in series.

So,

Probability that the system works is given by:

[1-{( 1 - p₁ p₂) (1 - p₃ p₄)}] p₅