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Math Help - Probability (complementary rule?)

  1. #1
    Newbie Almondzqueen's Avatar
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    Probability (complementary rule?)

    With one method of acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is rejected if there is at least one defect. An electronics company has just manufactured 2500 CDs, and 2% are defective. If 4 of the CDs are selected and tested, what is the probability that the entire batch will be rejected?

    Ok so I know how to set up the formula for this up to a point:
    P(reject batch)= P(at least one is defective)
    = 1-P (all good)
    =1-P (G1*G2*G3*G4)

    In the end the professor's answer is P(reject batch)=1-(2450/2500)*(2449/2499)*2448/2498)*(2447/2497)=.0777

    But I just don't understand HOW she got 2450 and so on....what did she subtract 2500 from to get that. I also understand why the denominator changed but don't get the numerator part of the final answer.
    Thanks!
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  2. #2
    Eater of Worlds
    galactus's Avatar
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    2% of the 2500 are defects. That means there are .02(2500)=50 defects in the batch.

    The probability the batch is rejected is if AT LEAST ONE is a defect.

    So, the easiest thing to do is use the complement and find the probability that none are defects and subtract from 1.

    Therefore, we have to choose 4 from the 2450 that are good and 0 from the 50 that are bad. We choose 4 from 2500 altogether.

    1-\frac{\binom{2450}{4}\cdot\binom{50}{0}}{\binom{25  00}{4}}
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