Consider a system of pipes with 5 electronically controlled valves as illustrate
ID: 3172694 • Letter: C
Question
Consider a system of pipes with 5 electronically controlled valves as illustrated in the diagram below. Each valve is currently closed. When activated, all valves should open. Suppose that upon activation, each valve has a 90% chance of opening, and each valve reacts independently from all other valves.
We seek to find the chance that there will be a clear path from Begin to End upon activation of the valves. In order to determine this probability, we determine the probability of each possible configuration of which valves open and which fail to open by completing the following table.
Fill in the remaining sections of the table on the following page.
Given there was a clear path from Begin to End, what is the conditional probability that valve #3 opened?
P( Value#3 opened | Clear path)= ______________
Explanation / Answer
Total probabilities sum for Yes= 0.99144
P( Value#3 opened | Clear path)= P(Value#3 opened and Clear path)/P(Clear Path)
=0.89667/0.99144 = 0.9044117647
Yes/NO # of open valves Probability Yes 5 (0.9^5)^(0.1^0) Yes 4 (0.9^4)^(0.1^1) Yes 4 (0.9^4)^(0.1^1) Yes 3 (0.9^3)^(0.1^2) Yes 4 (0.9^4)^(0.1^1) Yes 3 (0.9^3)^(0.1^2) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) Yes 4 (0.9^4)^(0.1^1) Yes 3 (0.9^3)^(0.1^2) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) No 2 (0.9^2)^(0.1^3) No 1 (0.9^1)^(0.1^4) Yes 4 (0.9^4)^(0.1^1) Yes 3 (0.9^3)^(0.1^2) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) No 2 (0.9^2)^(0.1^3) No 1 (0.9^1)^(0.1^4) Yes 3 (0.9^3)^(0.1^2) No 2 (0.9^2)^(0.1^3) No 2 (0.9^2)^(0.1^3) No 1 (0.9^1)^(0.1^4) No 2 (0.9^2)^(0.1^3) No 1 (0.9^1)^(0.1^4) No 1 (0.9^1)^(0.1^4) No 0 (0.9^0)^(0.1^5)