Ciara Rose

2022-07-23

Suppose 10 patients are to be tested for a blood disease and that the test in guaranteed to detect the disease. Furthermore, suppose that the probability that a patient has the disease is 0.02 and that presence of the disease in any patient is independent of all other patients. One option is to test all patients individually and that would require 10 tests. Another option is to combine blood samples from the 10 patients and test the combined sample. If the test is negative, then all patients are declared disease free and only 1 test is required. If, however, the combined sample tests positive, then all 10 patients are tested individually, requiring a total of 11 tests. In the latter option derive the probability distribution of , the number of tests, and determine Expected Value?
Among all the distribution models, it seems that geometric distribution seems best fit. But, when I use geometric distribution pmf and its' $EV=1/p$ , I get unreasonable answers.

Ali Harper

Expert

Step 1
Two possibilities: Case (A) no patients have the disease. The probability for this situation is ${p}_{A}=\left(1-0.02{\right)}^{n}$ and number of tests ${N}_{A}=1$. Case (B) at least one patient has the disease. Can you work out ${p}_{B}$? Hint: there is no other case.
Step 2
The expected value is $E\left[N\right]={p}_{A}{N}_{A}+{p}_{B}{N}_{B}$

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