# Quick statistics question

**URL:** <https://boards.straightdope.com/t/quick-statistics-question/67922>\
**Category:** Factual Questions\
**Created:** [June 7, 2001, 1:49pm UTC](https://boards.straightdope.com/t/quick-statistics-question/67922 "2001-06-07T13:49:37Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\
**Post date:** [June 7, 2001, 1:49pm UTC](https://boards.straightdope.com/t/quick-statistics-question/67922/1 "2001-06-07T13:49:37Z")

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I have 16 marbles, three of which are blue.

I am going to select three random marbles.

What is the percentage chance that at least one marble will be blue?

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**Author:** ![scr4](https://avatars.discourse-cdn.com/v4/letter/s/59ef9b/32.png) [@scr4](https://boards.straightdope.com/u/scr4)\
**Post date:** [June 7, 2001, 2:01pm UTC](https://boards.straightdope.com/t/quick-statistics-question/67922/2 "2001-06-07T14:01:52Z")

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It sounds like you pick 3 and keep them, rather than picking one, puting it back, picking another, etc. Is this correct?

The trick for this type of problem is to calculate the reverse case, i.e. the probability that none of the picked ones are blue. At first you have 13 non-blue out of 16, so the probability of picking a non-blue one is 13/16. For the next marble, you have 12 non-blue out of 15, and for the third, 11 out of 14. So the probability of all these happening is (13/16)_(12/15)_(11/14)=0.51. The probability of this **not** happening (i.e. at least one of the picked marbles is blue) is 1-0.51=0.49, or 49%.

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**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [June 7, 2001, 2:10pm UTC](https://boards.straightdope.com/t/quick-statistics-question/67922/3 "2001-06-07T14:10:36Z")

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Just in case you’re planning on putting them back in between…

In this case, the probability of picking no blue marbles in three tries is (13/16)[sup]3[/sup], which works out to be approximately .54. 1 - .54 = .46, which is the probability that at least one marble is blue.

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**Author:** ![Earl\_Snake-Hips\_Tucker](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/earl_snake-hips_tucker/32/3269_2.png) [@Earl\_Snake-Hips\_Tucker](https://boards.straightdope.com/u/Earl_Snake-Hips_Tucker)\
**Post date:** [June 7, 2001, 2:42pm UTC](https://boards.straightdope.com/t/quick-statistics-question/67922/4 "2001-06-07T14:42:42Z")

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Thanks a bunch for both scenarios.