A Binomial Problem

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    Can someone help to solve and explain how to solve this question

    A sampling plan involves selecting a sample of 50 from a very large lot, and rejecting the lot if 2 or more defects are found. If an incoming lot is 10% defective, what is the probability of rejecting the lot?

    A 0.6335
    B 0.9987
    C 0.5443
    D 0.9662


    Amit Kumar Ojha


    The correct answer is D(0.9662)

    Here is the solution….
    First of all consider this as a problem of Binomial Distribution. Our objective is to find the probability of getting 2 or more defects (thats when the lot will be rejected).
    Technically we can write it as P(X>=2)
    It is given in the problem that X is no. of defects, n=50, p=0.1, q=0.9

    P(X>=2) => 1- [P(X=0) + P(X=1)]

    P(X=0) => 50C0*(0.1^0)*(0.9^50) = 0.00515
    P(X=1) => 50C1*(0.1^1)*(0.9^49) = 0.02863

    Hence P(X>=2) => 1 – (0.00515 + 0.02863)
    => 0.966214

    Hence Option D is correct Answer

    Please let me know if you need any further clarification on this or any other such calculations.

    Good Luck!!!!


    Norbert Feher

    Or in minitab:

    1. Go to Graph/Probability distribution Plot!
    2. Select view probability
    3. Select Binomial distribution!
    Number of trials:50
    4. Select the shaded area tab
    5. Select X value instead of probability
    Right tail
    X value:2

    Result graph: shows a red shaded area of 0,9662

    So the previous answer was right :-)


    Amit Kumar Ojha

    Norbert shared a very simple way by which you can get the answer without getting into complexity of calculating the probabilities.

    In the above problem, since the calculations were not so tedious, hence you can do either way. However, in many real life scenarios the calculations are too complicated to be done manually.

    Hence it is good to use minitab, the way Norbert explained.

    Thanks Norbert :-))




    Thank you all for helping me solve the problem. Amit Kumar Your explanation is very easy to understand. This problem is from a CQE practice exam.

    I want to post the exam writer’s answer to this question. I had a hard time to understand his explanation although the answer is D.

    “Since the probability of success (a defect) is constant on each trial, the binomial distribution can be used to compute this probability. The probability of no successes in 50 trials, with the probability of success on a single trial of 0.1 is 0.0052. The probability of 1 success in 50 trials, with the probability of success on a single trial of 0.1 is 0.0286. The probability of 2 or more defects is 1-0.0052-0.0286=0.9662.”

    does the no successes means 0 defect?
    and 1 success means exactly 1 defect?

    what does the success on a single trial of 0.1 means?


    Amit Kumar Ojha

    Yes Tracy.

    You are right. No success in this case means 0 defects and 1 success implies 1 defect.
    Success on single trial of 0.1 means that if you test a sample of 100 units what is the probability of getting 10 defects.

    Hope I clarified your doubt :-))

    Best of Luck!!!



    Why are you using Binomial instead of Poisson?

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