The probability of the closing of the i th relay in the...

The probability of the closing of the $i$ th relay in the circuits shown in Figure 3.4 is given by $p_{i}, i=1,2,3,4,5$ If all relays function independently, what is the probability that a current flows between $A$ and $B$ for the respective circuits? Hint for $(b):$ Condition on whether relay 3 closes.

The probability of the closing of the $i$ th relay in the circuits shown in Figure 3.4 is given by $p_{i}, i=1,2,3,4,5$ If all relays function independently, what is the probability that a current flows between $A$ and $B$ for the respective circuits?
Hint for $(b):$ Condition on whether relay 3 closes.

Key Concepts

Series and Parallel Systems

In electrical circuits, components can be arranged in series or parallel, influencing the overall reliability. A series arrangement requires all components to work for the circuit to function, while a parallel arrangement allows for the circuit to function if at least one branch works. Recognizing these patterns is essential in evaluating the probability of current flow in complex systems.

Circuit Reliability Analysis

Circuit reliability analysis involves determining the probability that a circuit functions as intended, which may include current flow between specific points. This analysis takes into account the configuration of the circuit (such as series and parallel arrangements) as well as the individual probabilities of component success or failure.

Independent Events

In probability theory, events are considered independent if the occurrence of one does not affect the probability of occurrence of the other. In the context of circuits with relays, it means that whether one relay is closed or open does not influence the behavior of the other relays, allowing the overall system reliability to be computed as the product of the individual probabilities.

Conditional Probability

Conditional probability is the probability of an event given that another event has occurred. It is a powerful tool in analyzing complex systems by breaking down the problem into simpler cases. In circuit analysis, conditioning on whether a specific relay is closed can simplify the computation of the overall current flow probability.

Transcript

00:01 In problem 6 s xx we have a portion of a circuit with 5 relays the probability of closing of the ieth relay equals p i which means we have the probability 1 for closing relay 1 probability 2 here probability 3 probability 4 and finally we have probability 5 for closing all the 5 relays we want to calculate the probability that the current flows from point a to point b we can consider this as a portion of a circuit then we have at least one both from a to b the different both is we have both a which is 1 to 5 relay 1 then relay 2 then relay 5 the second bus b is relay 3 relay 4 relay 5 then to get the probability of part a we want to get the probability of a union b the probability of a union b is the probability of a the probability of a is the multiplication for their probabilities probability of 1 multiplied by relative 2 multiplied by probability of 5 because they act independently plus the probability of b as previous is the probability of 3 multiplied by 4 multiply by 5 minus the probability of the intersection between a and b the intersection is all relays are activated b1 b2 b3 b4 and b5 this is the answer for circuit number a for circuit b near we have fifth we have five relays and it has a different orientation from party to answer this we want to get the probability that the current flows from a to b let's see what is the buzzes that can part a the the current flows from a to b it can go from a to b from the first bus here one four this is a a one four the second bus b from 2 to 5 to 5 to 5 c is 234 2 3 4 and d is 1 3 5 the more direct way is to use the inclusion property which says that the current flows for the union of these buses a union union b, union c, union d.

03:30 As previous here, it's a union b.

03:34 Here, a, union b, union c, union d.

03:38 We can't get it using the inclusion.

03:42 It's the probability of a plus b plus c plus c plus d...

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