Stats - Independence (circuits)

In summary, the circuit will operate if there is a path of functional devices from left to right, and the probability of this happening is 0.929258. This can also be calculated using the formula P(A\cup B) = P(A) + P(B) - P(A\cap B).
  • #1
mattmns
1,128
6
Hello, my book has this question, and no examples (very) similar to it, so I am wondering if I did it correct :smile:
---------

The following circuit operates if and only if there is a path of functional devices from left to right. The probability that deach device functions is as shown. Assume that the probability that a device is functional does not depend on whether or not other devices are functional. What is the probability that the circuit operates?---|----0.9------0.8-------0.7-----|
---|-------------------------------|-----
---|----0.95----0.95------0.95-----|-----------------

So what I did was:

Let T denote the event that the entire top is true, meaning that everything functions on the top.
Let B denote the event that the entire bottom is true.

[itex]P(T \cup B) = 1 - P(T' \cap B')[/itex]
[itex]P(T' \cap B') = P(T')P(B')[/itex]
[itex]P(T') = 1 - P(T1 \cap T2 \cap T3) = 1 - P(T1)P(T2)P(T3)[/itex] T1 is the first thing on the top being functional (meaning 0.9), T2 being 0.8, etc
[itex]P(B') = 1 - P(B1 \cap B2 \cap B3) = 1 - P(B1)P(B2)P(B3)[/itex]
[itex]P(T') = 1 - (0.9)(0.8)(0.7) = 0.496[/itex]
[itex]P(B') = 1 - (.95)(.95)(.95) = 0.142625[/itex]
[itex]P(T' \cap B') = (0.496)(0.142625) = 0.070742[/itex]
[itex]P(T \cup B) = 1 - 0.070742 = 0.929258[/itex]
Which is my answer.

Does that look correct?
Thanks!
 
Last edited:
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  • #2
Reply

Hi! Your answer looks all right, but there is an alternative.

[tex] P(A\cup B) = P(A) + P(B) - P(A\cap B) [/tex]

So [tex]P(A) = (0.9)(0.8)(0.7)...
P(B) = (0.95)(0.95)(0.95)...
P(A\cap B) = (0.9)(0.8)(0.7)(0.95)(0.95)(0.95)... [/tex]

You will also obtain the same answer of 0.929258.
 
Last edited:

1. What is independence in circuits?

Independence in circuits refers to the property of two or more components in a circuit not influencing or affecting each other's behavior. In other words, the voltage or current in one component does not change due to the presence or absence of another component.

2. How is independence determined in a circuit?

Independence in a circuit can be determined by analyzing the circuit using Kirchhoff's laws and Ohm's law. If the voltage and current in each component can be calculated without taking into account the other components in the circuit, then the components are considered independent.

3. What are some examples of independent components in a circuit?

Some examples of independent components in a circuit are resistors, capacitors, and inductors. These components do not have any direct influence on each other's behavior and can be analyzed individually using Kirchhoff's laws and Ohm's law.

4. How does independence affect the overall behavior of a circuit?

Independence is an important property in circuits as it allows for easier analysis and understanding of the circuit. When components in a circuit are independent, their behavior can be analyzed separately and then combined to determine the overall behavior of the circuit.

5. What happens if components in a circuit are not independent?

If components in a circuit are not independent, their behavior can be affected by the presence or absence of other components. This can make the analysis of the circuit more complex and may require the use of more advanced circuit analysis techniques, such as nodal analysis or mesh analysis.

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