# statistics problem
**URL:** https://boards.straightdope.com/t/statistics-problem/565428
**Category:** Factual Questions
**Created:** [December 26, 2010, 3:22pm UTC](https://boards.straightdope.com/t/statistics-problem/565428 "2010-12-26T15:22:24Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@Carptracker](https://boards.straightdope.com/u/Carptracker)
#### Post date: [December 26, 2010, 3:22pm UTC](https://boards.straightdope.com/t/statistics-problem/565428/1 "2010-12-26T15:22:24Z")
I need to know the exact odds that, out of a population of 30,000, containing 100 articles that are “fake”, and from which a sample of 120 is tested (assuming testing is always correct and that the odds of pulling a fake sample is equivalent to the real ones) that a fake would be detected.
I’m sorry, I should know this stuff, but its been a long time and I am beginning to doubt my math.
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### Author:  [@Pasta](https://boards.straightdope.com/u/Pasta)
#### Post date: [December 26, 2010, 3:51pm UTC](https://boards.straightdope.com/t/statistics-problem/565428/2 "2010-12-26T15:51:11Z")
First calculate the odds that no fake will be detected:
probability of seeing no fake = P(first draw is real) \* P(second draw is real given that first draw is real ) \* … \* P(120th draw is real given that all preceding draws are real)
probability of seeing no fake = (29900/30000) \* (28999/29999) \* … \* (29781/29881)
probability of seeing no fake = 0.669
So, the probability of seeing at least one fake is one minus this:
P(see a fake) = 1 - 0.669
P(see a fake) = 0.331
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### Author:  [@Pasta](https://boards.straightdope.com/u/Pasta)
#### Post date: [December 26, 2010, 4:01pm UTC](https://boards.straightdope.com/t/statistics-problem/565428/3 "2010-12-26T16:01:36Z")
And [a graph](http://www.flickr.com/photos/53143438@N04/5293638204/).
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### Author:  [@Carptracker](https://boards.straightdope.com/u/Carptracker)
#### Post date: [December 26, 2010, 4:20pm UTC](https://boards.straightdope.com/t/statistics-problem/565428/4 "2010-12-26T16:20:33Z")
Excellent. Thank you. I _was_ doing it wrong, forgetting to take out the influence of the real ones I removed searching for the fakes. It makes a small difference.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [December 26, 2010, 8:51pm UTC](https://boards.straightdope.com/t/statistics-problem/565428/5 "2010-12-26T20:51:58Z")
You didn’t specify whether you’re sampling with or without replacement, and **Pasta** ’s answer assumes that you’re sampling without. In that case, the number of fake articles you pull is distributed [hypergeometrically](http://en.wikipedia.org/wiki/Hypergeometric_distribution) with N = 30000, m = 100 and n = 120.