When turned on, each of the three switches attached in the circuit diagram below

Question

When turned on, each of the three switches attached in the circuit diagram below works properly with probability 0.9. If a switch is working properly, current can flow through it when it is

turned on. Let Y be a random variable defined as the number of closed paths from a to b

when all three switches are turned on. Find the probability mass function for Y.

When turned on, each of the three switches attached in the circuit diagram below works properly with probability 0.9. If a switch is working properly, current can flow through it when it is turned on. Let Y be a random variable defined as the number of closed paths from a to b when all three switches are turned on. Find the probability mass function for Y.

Explanation / Answer

To determine the pmf, look at the problem this way:
There are three possibilities:

0 Paths, 1 Path, 2 Paths.

For there to be 0 paths:
Switch #3 and either switch #1 or #2 have to be opened.

Note: (probability of open = 1-0.9 = 0.1)
Also note: Probability of A or B is P(A) + P(B) (Given they are independent)

This means the probability of that is:

P(0 Paths) = P(#3 opened) * (P(#1 Opened) + P(#2 Opened)),

= (0.1*(0.1+0.1))=0.02

For there to be 2 paths, all the switches have to be closed. This means that the probability has to be:

P(2 paths) =

P(#1 closed) * P(#2 closed) * P(#3 Closed) = 0.9*0.9*0.9= 0.729.

There is only one option after "0 paths" and "2 paths" have been determined. That is "1 Path".

Since the total probability has to equal 1,

P(1 path) = 1- P(2 paths) - P(0 paths) = 1-0.729-0.02 = 0.251

So The solution is:
P(Y<0) = 0

P(Y=0) = 0.02

P(Y=1) = 0.251

P(Y=2) = 0.729

P(Y>2) = 0

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