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# The Internal Revenue Service estimates that 8% of all taxpayers filling out long forms make mistakes.(a) An IR?

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The Internal Revenue Service estimates that 8% of all taxpayers filling out long forms make mistakes.(a) An IRS employee starts to randomly select forms—one at a time—to check for mistakes. What is the probability that the first form with mistakes is the 7th one she checks?(b) The same IRS employee announces at lunch one day that she checked 25 forms this morning and didn’t find mistakes on any of them. Is this surprising enough so that her supervisor should worry about whether she is missing errors? Explain.

### 2 Answers

- J. J..Lv 78 years agoFavorite Answer
A. For the first mistake to appear on the 7th form checked it means that 7 individual events must all occur. The first six must all be correct and the 7th must be wrong.

If the probability of an error on the form is 8% it means that the probability of a correct form is 92%

Therefore the probability that the first mistake is on the 7th form =

0.92 * 0.92 * 0.92 * 0.92 * 0.92 * 0.92 * 0.08

= 0.048508 or 4.85% to two places of decimals

B. No it is not a sufficiently extreme situation to worry about (YET!)

The probability of getting 25 out of 25 correct is 0.92^25

This equates to 0.124364 or to put it in other words approximately once in every 8 occasions

ie in every 8 batches of 25 forms there should be one batch where there are no mistakes

Increasing the number in the batch to 50 will reduce the liklihood to 0.92^50 or approx once every 64 batches of 50 forms. At this level the supervisor should begin to worry and check her work, but at one in eight, it is not unreasonable to find this occurance.

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- JeffLv 68 years ago
8% means that every 12.5 files will have an error or 2 out of 25. On average the worker would have 2 mistakes that morning. Since you expect 23 perfect files getting 2 more isn't a big deal or that unlikely to happen. If it was 0 mistakes out of 250 then there is something wrong, but with a sample size this small it is hard to tell.

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