Question

Using the data found in Table 4 and Bayes’ Formula, determine the probability that a randomly selected patient will have Strep Throat given the SARTD test result was positive. Use the CDC stated prevalence of 25%. Round answer to nearest hundredth of a percent (i.e. 45.67%).

Then using the same Table 4, and Bayes’ Formula, determine the probability that a randomly selected patient will not have Strep Throat given the SARTD test result was negative. Use the CDC stated prevalence of 25%. Round answer to nearest hundredth of a percent.

	Strep Pos	Strep Neg	Total
SARTD Pos	80	23	103
SARTD Neg	38	349	387
Total	118	372	490
Table 4: SARTD vs conventional culture

Answer 1

Let us consider two events A and B, where

A: a randomly selected patient will have Strep Throat

B: the SARTD test result of a randomly selected patient is positive

Then,

A^c: a randomly selected patient will not have Strep Throat

B^c: the SARTD test result of a randomly selected patient is negative

Thus, we need to find P(A|B) and P(A^c|B^c)

Now, the result of the SARTD test for a patient depends on whether he/she has Strep Throat.

Thus, event A (or A^c) corresponds to having (or not having) the disease and event B (or B^c) corresponds to the result of having the disease being positive (or negative).

Hence, event A may be looked upon as the cause of event B. The required events correspond to the cause given the effect has occurred. Thus, to find the probabilities of the required events , we need to apply Bayes' Theorem.

Now, from the given data, we have:

and, similarly,

Thus, from Bayes' Theorem, we have,

and

Hence, we have