How many cows are required per farm so as the probability of getting at least one diseased cow is 70% (acceptable level to us, using area under ROC cutoff point), given that the disease prevalence is 25%?

My solution for this problem is by using binomial distribution. The following is the formula for the distribution,

In our context,

*x*, number of success

*n*, sample size

*p*, prevalence

and to solve our problem

*p*(1 or more) = 1 -

*p*(0)

Using spreadsheet, we can find the value of

*n*iteratively (I prefer LibreOffice Calc). Just key in the following function (or just put up the formula above)

=1 - BINOMDIST(n, x, p, 1)

thus in our context

=1 - BINOMDIST(n, 0, 0.25, 1)

after playing around with

*n*, I found

for

*n*= 4,

*p*(1 or more) = 1 - 0.316

*=*0.684

for

*n*= 5,

*p*(1 or more) = 1 - 0.237

*=*0.763

I'd go for 5 cows...

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