# Probability of strictly guessing on a multiple choice test?

**URL:** <https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849>\
**Category:** Factual Questions\
**Created:** [February 21, 2010, 4:47am UTC](https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849 "2010-02-21T04:47:27Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\
**Post date:** [February 21, 2010, 4:47am UTC](https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849/1 "2010-02-21T04:47:27Z")

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Someone posted this question on another message board (but I know THIS is the place to get an answer). 😃  
Anyway, if someone were just to make outright guesses on a 60 question multiple choice test (5 choices per question), what is the probability of getting 40 correct?

At first I thought this might be similar to a lottery probability in which you can calculate ‘k’ ways of taking ‘n’ things but the 5 choices per question makes it a different type of problem.

Is this an occupancy problem? (Random objects being dropped into urns).

Anyway, since I have asked (and answered) a few probability questions here in the past, this has really got me intrigued. Does anyone here know how to figure out the probability?

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**Author:** ![AK84](https://avatars.discourse-cdn.com/v4/letter/a/85e7bf/32.png) [@AK84](https://boards.straightdope.com/u/AK84)\
**Post date:** [February 21, 2010, 5:08am UTC](https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849/2 "2010-02-21T05:08:24Z")

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Are you saying that every question is a “guess”? Because in most MCQ question which I have taken you can usually eliminate at least two which are not right and and then guess between three or two.

Would change the probabilty.

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**Author:** ![ragerdude](https://avatars.discourse-cdn.com/v4/letter/r/d07c76/32.png) [@ragerdude](https://boards.straightdope.com/u/ragerdude)\
**Post date:** [February 21, 2010, 5:14am UTC](https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849/3 "2010-02-21T05:14:48Z")

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This is a “Bernoulli trial” problem.

Is your question “what is the probability of getting EXACTLY 40 correct?” or “what is the probability of getting AT LEAST 40 correct?”, because those are different questions.

Either way, both are very small probabilities.

Using this, with p = .2, n = 60, and x = 40:  
[http://stattrek.com/Tables/Binomial.aspx](http://stattrek.com/Tables/Binomial.aspx)

I get the following. Probability of “exactly 40” is 5.314e-15 and the probability of “at least 40” is 5.995e-15.

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<div class="post-metadata">

**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\
**Post date:** [February 21, 2010, 6:23am UTC](https://boards.straightdope.com/t/probability-of-strictly-guessing-on-a-multiple-choice-test/529849/4 "2010-02-21T06:23:01Z")

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**AK84**  
Yes, each answer is an outright guess. No attempt has been made to increase the probability from .2 to anything higher.  
And yes I know precisely what you mean about the typical multiple choice test question.  
Mount Rushmore is located in which US State?  
A) Hawaii  
B) Florida  
C) Louisiana  
D) North Dakota  
E) South Dakota

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**ragerdude**  
Thanks for your answer. (I had a feeling that the probability would be incredibly miniscule).  
I appreciate the time you spent on that and now I know that it is a “Bernoulli Trial” probability problem.  
That is one heck of an online calulator you found - I’m definitely bookmarking it.