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Math Help - Probability problem

  1. #1
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    Probability problem

    The probabilities that machines X, Y, Z will be performing well are \frac{1}{3},\frac{1}{4},\frac{1}{5}. Find the probability that at most two of the machines will be operating.

    P(at most two operating)
    =1-P(all operating)
    = \frac{59}{60}

    This is found in my math book. Can anyone explain why we need to use 1-P(all operating)? and I found out this 1-P(all operating) cannot be used when finding P(all machines not working)... Why?
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  2. #2
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    Quote Originally Posted by cloud5 View Post
    The probabilities that machines X, Y, Z will be performing well are \frac{1}{3},\frac{1}{4},\frac{1}{5}. Find the probability that at most two of the machines will be operating.

    P(at most two operating)
    =1-P(all operating)
    = \frac{59}{60}

    This is found in my math book. Can anyone explain why we need to use 1-P(all operating)? and I found out this 1-P(all operating) cannot be used when finding P(all machines not working)... Why?
    Consider events A,B defined as: A = (at most 2 functioning) and B = (all 3 functioning). A and B are complements of each other. In other words, B means: not A. So P(A) + P(B) = 1.

    P(all machines not working) is (1-1/3)(1-1/4)(1-1/5). Do you see why?

    Edit: Originally I used X,Y in place of A,B not noticing that this choice of letters conflicted with the machine names.
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  3. #3
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    I understand now. Thank you.

    Quote Originally Posted by undefined View Post
    Consider events A,B defined as: A = (at most 2 functioning) and B = (all 3 functioning). A and B are complements of each other. In other words, B means: not A. So P(A) + P(B) = 1.

    P(all machines not working) is (1-1/3)(1-1/4)(1-1/5). Do you see why?

    Edit: Originally I used X,Y in place of A,B not noticing that this choice of letters conflicted with the machine names.
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