# Probability that current could pass through

• pp123123
In summary: The final expression should be p^5-5p^4+2p^3+2p^2. Your friends' expression may have been simplified by assuming p=1, which would give a probability of 0 for current passing.
pp123123

## Homework Statement

http://imageshack.com/a/img546/5686/nqpn.jpg
I am looking for the probability of the current passing from the left to right given that there are 5 independent switches of probability p. However I cannot figure out the correct solution.

## The Attempt at a Solution

Case I: Assume Switch E is closed so I have a probability of
P((AB) or (CD))=P(AB)+P(CD)-P(AB and CD)=2p^2-p^4

Case II: Assume Switch E is opened so I have a probability of
P((A or C) and (B or D))=(P(A)+P(C)-P(AC))(P(B)+P(D)-P(BD))=(2p-p^2)^2

so overall probability is (1-p)(2p^2-p^4)+p(2p-p^2)^2 which could be simplified to 2p^5-5p^4+2p^3+2p^2

however some friends of mine told be that it should be p^5-5p^4+2p^3+2p^2. I hope to know what's wrong in my calculation. Thanks!

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I got the same answer you did by enumerating all 32 possibilities, identifying which ones belong in the event, and adding up the corresponding probabilities.

You should also consider what the probability should be if p=1. Your friends' expression says the probability is 0. Does that make sense?

pp123123 said:

## Homework Statement

http://imageshack.com/a/img546/5686/nqpn.jpg
I am looking for the probability of the current passing from the left to right given that there are 5 independent switches of probability p. However I cannot figure out the correct solution.

## The Attempt at a Solution

Case I: Assume Switch E is closed so I have a probability of
P((AB) or (CD))=P(AB)+P(CD)-P(AB and CD)=2p^2-p^4

Case II: Assume Switch E is opened so I have a probability of
P((A or C) and (B or D))=(P(A)+P(C)-P(AC))(P(B)+P(D)-P(BD))=(2p-p^2)^2

so overall probability is (1-p)(2p^2-p^4)+p(2p-p^2)^2 which could be simplified to 2p^5-5p^4+2p^3+2p^2

however some friends of mine told be that it should be p^5-5p^4+2p^3+2p^2. I hope to know what's wrong in my calculation. Thanks!

You have not said whether p is the probability that a switch is open (blocking current) or closed (allowing current). I will assume p = probability the switch is closed, allowing current.

There are only 4 paths (routes) from start S to finish F, namely: R1 = ab, R2 = aed, R3 = cd, R4 = ceb. Current can go from S to F if at least one of R1, R2, R3, R4 is 'closed' (meaning it allows current). The probability that R1 is closed is P(R1) = p^2, since both a and b must be closed. You can get the other route probabilities in a similar way. Then, all you need to do is figure out how to find the probability that at least one route is closed, given all the individual route probabilities, etc.

Last edited by a moderator:
pp123123 said:
. I hope to know what's wrong in my calculation. Thanks!

Your calculation is correct, except that case 1 is really for switch E open, and case 2 for switch E closed, instead of the other way around.

## What is probability?

Probability is the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

## How is probability calculated?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if a coin is tossed and there are 2 possible outcomes (heads or tails) and only 1 favorable outcome (heads), the probability would be 1/2 or 0.5.

## What factors affect the probability of current passing through?

The probability of current passing through is affected by several factors, including the resistance of the material through which the current is passing, the voltage applied, and the temperature of the material.

## How can probability be used in practical applications?

Probability is used in many practical applications, such as predicting the likelihood of a certain event occurring, making decisions based on risk assessment, and analyzing data in fields such as economics, medicine, and engineering.

## What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and predicts the likelihood of an event occurring. Experimental probability is based on actual data and results from conducting an experiment or observing real-life events. As more data is collected, the experimental probability may approach the theoretical probability.

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