In some cases, instead of complementary events B and B′, we have complementary events B1​, B2​ and B3​. For example, instead of

• B: "has COVID" & B′: "does not have COVID"

We might have

• B1​: "has COVID", B2​: "has flu", B3​: "has neither"

In this case Bayes' theorem can be generalized to

Example

A patient presenting with a fever is believed to have either COVID or the flu. It is given that 80% of COVID patients and 95% of flu patients have fevers. 10% of patients who don't have COVID or the flu present with a fever. It is known that 2% of people to have COVID and 3% have the flu at any given time. Find the probability that the patient has COVID.

Let

• A be the event "patient has fever"

• B1​ "patient has COVID"

• B2​ "patient has the flu" and

• B3​ "patient has neither"

From the given information we have P(B1​)=0.02, P(B2​)=0.03 and therefore P(B3​)=1−0.02−0.03=0.95.

We also know P(A∣B1​)=0.8, P(A∣B2​)=0.95 and P(A∣B3​)=0.1. Thus

Or an 11.5% chance.