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Question

A company manufacturers video games with a current defect rate of 0.95%. To make sure as few defective video games are delivered as possible, they are tested before delivery. The test is 98% accurate at determining if a video game is defective. If 100000 products are manufactured and delivered in a month, approximately how many defective products are expected to be delivered?


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Solution

Finding the number of defective products:

Step-1: Construction of a Binomial Random Variable:

For each piece of product delivered, there are only two possible outcomes: either it is defective or it is not.

The probability of a piece of product delivered being defective is independent of any other pieces, which means that the binomial distribution is used to solve this problem.

Let X be the random variable that denotes the number of defective product delivered. Here, we have to find expectation of X, EX.

Step-2: Finding the expectation:

Formula to be used: We know that for a binomial random variable X with probability of success p and with n trials, the expectation is EX=np.

Here, n=100000.

Given that 0.95% of the products are defective and the accuracy of the test before delivery is 98%. So, (100-98)=2% of the defective products are delivered. Hence, we must have:

p=0.95%×2%=0.95100×2100=1.910000

Hence,

EX=np=100000×1.910000=19

Therefore, approximately 19 defective products are expected to be delivered.


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