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I know how to apply Bayes Theorem to determine the likelihood a patient has a given disease if multiple trials return a positive result. In this case, if two trials both return positive:

$$ P(\text{User}\mid++) = \frac{P(\text{User})P(+\mid\text{User})^2}{P(\text{User})P(+\mid\text{User})^2+(1 − P(\text{User})) P(+\mid\neg\text{User})^2} $$

However, how could I modify the above formula if one conducts three trials, and exactly two tests come out positive? What would be the likelihood that the patient has the disease then?

I would assume that $_3C_2$ comes into play as well, but I'm lost as to how to incorporate the negative result into the probability calculation.

Michael Hardy
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mitmath514
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  • So "User" means a patient who has a given disease? – Michael Hardy Sep 18 '18 at 00:28
  • Draw the tree diagram. It should have 4 trials (User/Non-User ; +/- ; +/- ; +/-) and 16 outcomes. 6 of the outcomes pertain to the event that exactly two tests of three return positive results; 3 of these 6 outcomes also pertain to the patient being a User/Unhealthy/Diseased. The required probability P(User | exactly two tests of three return positive results) is then [sum of the probabilities of the latter 3 outcomes]/[sum of the probabilities of the former 6 outcomes], exactly as given by Graham. For reference, a similar tree diagram is here. – ryang Feb 21 '22 at 16:32

1 Answers1

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I would assume that $^3\mathrm C_2$ comes into play as well,

Yes and no. They cancel.

but I'm lost as to how to incorporate the negative result into the probability calculation.

Just include them.

To shorten typesetting, let $U$ be the event of the subject being a user (???), $S$ be the count of positive test results among three tests (which are presumed to be conditionally iid for a given subject, each with conditional probabilities $P(T\mid U)$ and $P(T\mid \neg U)$ ).

$$\begin{align}\mathsf P(U\mid S=2) &=\dfrac{\mathsf P(S=2\mid U)~\mathsf P(U)}{\mathsf P(S=2\mid U)~\mathsf P(U)+\mathsf P(S=2\mid \neg U)~\mathsf P(\neg U)}\\[1ex]&= \dfrac{^3\mathrm C_2\mathsf P(T\mid U)^2~\mathsf P(\neg T\mid U)~\mathsf P(U)}{^3\mathrm C_2\mathsf P(T\mid U)^2~\mathsf P(\neg T\mid U)~\mathsf P(U)+^3\mathrm C_2\mathsf P(T\mid \neg U)^2~\mathsf P(\neg T\mid\neg U)~\mathsf P(\neg U)}\\[1ex]&= \dfrac{\mathsf P(T\mid U)^2(1-\mathsf P(T\mid U))~\mathsf P(U)}{\mathsf P(T\mid U)^2(1-\mathsf P(T\mid U))~\mathsf P(U)+\mathsf P(T\mid \neg U)^2(1-\mathsf P(T\mid\neg U))(1-\mathsf P(\neg U))}\end{align}$$

Graham Kemp
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