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title: "Binomial method of estimating a probability - not recommended"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575355/Binomial%20method%20of%20estimating%20a%20probability%20-%20not%20recommended"
format: markdown
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![image](media://6aa07293-cbbb-4911-8051-ff145acf6dfc)

  


> Macro (mathblock)

  


<span style="color: #000000">We assume that each measurement point is a binomial random variable that has a probability </span><span style="color: #000000">*p*</span><span style="color: #000000"> of having the characteristic of interest. If all measurements are independent, and we assign a value to the measurement of 1 when the measurement has the characteristic of interest and 0 when it does not, the measurements can be thought of as a set of Bernoulli trials. Letting </span><span style="color: #000000">*P*</span><span style="color: #000000"> be the random variable of the proportion of </span><span style="color: #000000">*n*</span><span style="color: #000000"> of this set of trials {</span><span style="color: #000000">*X*</span><sub><span style="color: #000000">*i*</span></sub><span style="color: #000000">} that have the characteristic of interest, it will take a distribution given by:</span>

![image](media://e3e0bff4-2e73-48c6-a2d6-41d1dcedb3f6)

  


  


> Macro (mathblock)

                                            (1)

  


We observe *s* of the *n* trials with the characteristic of interest, so *s*/*n* is our one observation from the random variable *P* which is also our [maximum likelihood](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575336/), and [unbiased](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26586655), estimate for *p*. Switching around Equation 1, we can get an uncertainty distribution for the true value of *p*:

![image](media://818f4ef5-e843-418c-91a1-99e1bdfd1659)

  


  


> Macro (mathblock)

                                           (2)

  


This exactly equates to the non-parametric and parametric [Bootstrap estimates of a Binomial probability](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575408/). Equation 2 is awkward since it will only allow (*n*+1) discrete values for *p* i.e. {0, 1/*n*, 2/*n*, …, 1/(*n*-1), 1}, whereas our uncertainty about *p* should really take into account all values between zero and 1:

  


![image](media://f03586b0-188c-4a69-abbf-202129fbe65c)

<span style="color: #000000">**Figure 1**</span><span style="color: #000000">: Example of Equation 2 estimate of p where </span><span style="color: #000000">*s*</span><span style="color: #000000"> = 5, </span><span style="color: #000000">*n*</span><span style="color: #000000"> = 10</span>

  


  


It also makes no sense that p could be either zero or one, of course.

  


  


  


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