# Probability of a binomial distributed variable

**URL:** <https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977>\
**Category:** Questions\
**Tags:** statistics-and-probability\
**Created:** [January 10, 2025, 7:25am UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977 "2025-01-10T07:25:27Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![joachimfroehlich](https://avatars.discourse-cdn.com/v4/letter/j/7c8e57/32.png) [@joachimfroehlich](https://community.wolfram.com/u/joachimfroehlich)\
**Post date:** [January 10, 2025, 7:25am UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/1 "2025-01-10T07:25:27Z")

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Dear community,  
if I enter  
probability 0 \<= x \<= 199 if x is binomial with n = 629 and p = 1/3  
I have no result.  
If I enter  
probability 0 \<= x \<= 199 if x is binomial with n = 629 and p = 0,3333333333  
I got a result.  
Any inspiration, what I did wrong?  
Regards.

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**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [January 10, 2025, 11:34am UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/2 "2025-01-10T11:34:02Z")

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I do get a result with

```
Probability[0 <= x <= 199, 
 Distributed[x, BinomialDistribution[629, 1/3]]]

```

It is a rather complicated rational number. I can apply `N` to get a floating point approximation.

Maybe you had a semicolon `;` at the end of the expression? That suppresses the display of the output.

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<div class="post-metadata">

**Author:** ![joachimfroehlich](https://avatars.discourse-cdn.com/v4/letter/j/7c8e57/32.png) [@joachimfroehlich](https://community.wolfram.com/u/joachimfroehlich)\
**Post date:** [January 10, 2025, 12:30pm UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/3 "2025-01-10T12:30:36Z")

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Thanks Gianluca,  
no, I haven’t put a semicolon at the end.  
I tried to solve the Problem wit my handy  
and it works.  
Best regards  
Joachim

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**Author:** ![danl](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/danl/32/41267_2.png) [@danl](https://community.wolfram.com/u/danl)\
**Post date:** [January 10, 2025, 5:00pm UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/4 "2025-01-10T17:00:33Z")

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Or maybe the exact value computation timed out on Wolfram|Alpha but the numeric one did not?

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**Author:** ![tomasgarza10](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/tomasgarza10/32/14613_2.png) [@tomasgarza10](https://community.wolfram.com/u/tomasgarza10)\
**Post date:** [January 10, 2025, 6:15pm UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/5 "2025-01-10T18:15:09Z")

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I obtain the result in less than 6 seconds (rather long, but not really complicated)

probability 0 \<= x \<= 199 if x is binomial with n = 629 and p = 1/3

2790202042683170027396338572156271486236764921820554406324053343438420  
7473524012382076372430640929027629909848334225903726967528396496546094  
2768804990201714967763531874391873967728927521481203010698948463470386  
5556684010030811850825222228971811003257372984250289096178370247660188  
8824446582320529408/  
1428981838579134870963576158997950040247615937024229320041839443209485  
7063291801552419719892590341365005232809905728568571532045291896518962  
0416249357065663843887173052694852955236403957825594931793818273072298  
1380571684806205514443614319046207595729139902191275491143892128100897  
71517714979850768987

In[3]:= N[%]

Out[3]= 0.195258

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<div class="post-metadata">

**Author:** ![joachimfroehlich](https://avatars.discourse-cdn.com/v4/letter/j/7c8e57/32.png) [@joachimfroehlich](https://community.wolfram.com/u/joachimfroehlich)\
**Post date:** [January 10, 2025, 6:37pm UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/6 "2025-01-10T18:37:37Z")

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Hello again,  
I used the alpha widget: Berechnen von Werten der Binomialverteilung. No result.  
Regards.

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<div class="post-metadata">

**Author:** ![tomasgarza10](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/tomasgarza10/32/14613_2.png) [@tomasgarza10](https://community.wolfram.com/u/tomasgarza10)\
**Post date:** [January 11, 2025, 1:02am UTC](https://community.wolfram.com/t/probability-of-a-binomial-distributed-variable/24977/7 "2025-01-11T01:02:46Z")

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Incidentally, the normal approximation to the binomial gives the value 0.183472 which, compared to the 0.195258 obtained directly, has a relative error of 6.4%. Too large, maybe…, depending, of course, on the purpose of the exercise. But, of course, it takes almost the same time and effort to calculate both results. So, come to think of it, nowadays the central limit theorem is perhaps as outdated as the tables of logarithms (for numerical approximations purposes).