the probability that each of 2 component works is 2/3, what is the probability that the machine works?

Question

QUESTION: A machine is made up of 2 components, A and B. Each component either works or fails. Failure or nonfailure of one component is independent of the failure or nonfailure of the other component. The machine works if at least one of the components works. If the probability that each component works is 2/3, what is the probability that the machine works?

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I know that the 2 components' probability of working are independent events, my approach to this was multiplying their probabilities to each other: (2/3)*(2/3) but that seems to be wrong. Can someone please explain why its wrong? In other words, what would the question be for my approach described previously to be correct?

The answer key says so take the probability that each component doesnt work which is 1 - (2/3) = 1/3 and multiply them to each other: (1/3)*(1/3) = 1/9 Now, this fraction is the probability that the machine doesnt work so the complement of this fraction is the probability that the machine works.

Answer 1

Your approach only considers the possibility that both A & B will work. You neglected the possibility that one of the two will fail:



P(A works) =2/3

P (A fails) =1/3

P(B works) =2/3

P(B fails) = 1/3



There are four combinations:



A works/ B works

A works/ B fails

A fails/ B works

A fails/ B fails



The respective probabilities are:



P (A works/ B works) =(2/3) *(2/3) = 4/9

P (A works/ B fails) = (2/3) * (1/3) = 2/9

P (A fails/ B works) = (1/3)* (2/3) = 2/9

P (A fails/ B fails) = (1/3) *(1/3) =1/9



The sum of the probabilities is one. That is, there is a 100% probability that one of the four combinations will occur.

Only the last one (both A and B fail) would cause the machine to fail.

Answer 2

The answer key explanation is correct. If the probability of an event is P, then the probability of that event NOT happening is

1 - P



If the probability of each component working is 2/3, the probability of it not working is

1 - 2/3 = 1/3.



The probability of both components failing (and therefore the probability that the machine fails) is

1/3 * 1/3 = 1/9.



The probability that the machine does not fail is therefore

1 - 1/9 = 8/9.



The key is in the line: "The machine works if at least one of the components works."



If both components had to work for the machine to work, the probability of the machine working would have been

2/3 * 2/3 = 4/9.

But in this case, we needed only that at least one of them worked. That's why your approach was wrong.

Answer 3

Multiplying 2/3 by 2/3 would give the probability of the machine working if both components had to work.

Answer 4

Multiplying the probabilities that both will work will tell you just the number of times both components work. However only one component has to work for the machine to work so it is correct that you should then look at the times that both components will fail at the same time.



A simple way to look at this - draw a 3x3 grid, with either a tick or cross for each component in each cell. Count the number of cells that have at least one tick.