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We assume that each measurement point is a binomial random variable that has a probability p 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 P be the random variable of the proportion of n of this set of trials {Xi} that have the characteristic of interest, it will take a distribution given by:



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, and unbiased, estimate for p. Switching around Equation 1, we can get an uncertainty distribution for the true value of p:



This exactly equates to the non-parametric and parametric Bootstrap estimates of a Binomial probability. 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:

Figure 1: Example of Equation 2 estimate of p where s = 5, n = 10

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

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