Probability of current flow through circuit

[snoggerT]

The probability of the closing of the ith relay in the circuits shown is given by p_i. Let p_1 = 0.9, p_2 = 0.6, p_3 = 0.1, p_4 = 0.7, p_5 = 0.2. If all relays function independently. what is the probability that a current flows between A and B for the respective circuits?

https://webwork.math.lsu.edu/webwork2_course_files/Math-3355-01-Fall2009/tmp/gif/HW4-prob1-ur_pb_4_14.gif

The Attempt at a Solution

I am working on the circuit in part B. I broke down the problem into 4 events in which the current could flow through the circuit. I labeled them A, B, C and D. I know that the problem is just P(AUBUCUD), but I do not know how to do a union for 4 events. Can someone please help?

 

[LCKurtz]

You need the inclusion-exclusion principle:

http://en.wikipedia.org/wiki/Inclusion-exclusion_principle

and scroll down the "In Probability" section.

 

[snoggerT]

You need the inclusion-exclusion principle:

http://en.wikipedia.org/wiki/Inclusion-exclusion_principle

and scroll down the "In Probability" section.

- That is what I tried using actually, but I guess I'm not quite sure how to set up the formula once you get past 3 events.

 

[LCKurtz]

Try just using a chain of conditional probabilities. Let:
C = event that current flows
s1 = event switch 1 is closed
s2 = event switch 2 is closed etc.

So you could start like this:

P(C) = P(C|s1)P(s1) + P(C|s2)P(s2) = P(C|s1)(.9) + P(C|s2)(.6)

Now to calculate P(C|s1) condition it on s3 and s4 and fill in what you know. Keep going like that.