Assume that a diagnostic test for EPM (equine protozoal myoeloencephalitis) has a sensitivity of 89% and a specifity of 92%. The prevalence of the disease is 1%

  1. What is the probability that a randomly selected horse will test positive on the diagnostic test if the horse is healthy?
  2. What is the probability that a randomly selected horse (from the total population) will test positive on the diagnostic test?

I think in 1. I have to use the definition of conditional probability (https://en.wikipedia.org/wiki/Conditional_probability the one that says Kolmogorov definition) to calculate the probability of the intersection (positive and healthy), but I'm a bit stumped on 2.

edit: I think I might have figured it out. I have to use the law of total probability, it seems. Still, I'd appreciate some feedback if possible, since I have nowhere to check if my answer is correct and I'm very new to this stuff.


(second question only):

We want to find out the positive fraction. In other words, we need to find out the percentage of true positives ($TP$) and false positives ($FP$) and add them up.

We know that $Sensitivity =\frac{TP}{Disease (D)}=\frac{TP}{TP+FN}$, and

$Specificity=\frac{TN}{No\,Disease\,(\bar D)}=\frac{TN}{TN+FP}$.

The sensitivity give you the $TP\, ratio$. The question boils down to realizing that the $FP\,ratio$ is just $1\,-\,Specificity$. Here's why: $FP\,ratio = \frac{FP}{\bar D}=\frac{FP}{TN+FP}=1-\frac{TN}{TN+FP}$.

Now we know that we want:

$p\,(+)=p(TP\cup FP) =\,p\,(TP)\,+\,p\,(FP)=TP\,ratio+FP\,ratio$.

There is one slippery point to this question, which is that it is not asking us for the probability of disease being present, given that a test turned out positive. This would have entailed using Bayes theorem and knowing the prevalence of the disease.


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