Question

Two doctors independently examine a person randomly chosen from a certain population to check for the presence or absence of a particular disease. Let CA be the event that Doctor A makes the correct diagnosis, CB the event that Doctor B makes the correct diagnosis, and D the event that the randomly chosen patient actually has the disease in question. The doctors are equally skilled, so we have P(CAD) = P(CB|D) = pi and P(CA|DC) = P(CB|DC) = po Finally, let 0 = P(D) be the prevalence of the disease in the population, that is, the probability the patient has the disease. (a) Prove that the probability both doctors make the correct diagnosis is given by 0(pî - p.) + pa. (b) Show that, in general, P(CB|CA) P(Cb). What do you conclude? How would you explain this to a non-statistician? (c) Find all the possible values of 0, po, and pı for which we have P(CBCA) = P(Cb). What special cases do these correspond to?

Answer 1

a) P ( both diagnoise + P(both diagnosid anrectly ID") PCO) encally) =P(Conca) < PC both diagonise correctly I pistemed) PCD) PCCA1D) PCCBIP) PCD) + PCCAID!) PCCB 10% P(D) piro + 9% (1-0) = 0 (95 - 96) + P " 11 = b) PC CAICA) = 9(ConCA) PCCA) & acp,%-por) + por PCCA/DP603 + PCCAT DUPCOCI OCP, 2-po) thor PCCB) o. (bi- po :) + po ochi-Po) + Po CB & CA are . PCPBICA) EP CCB) → dependent. Honce the correct diagononis of likelihood at cooreet diagnosis decoro at B does a depend upon the correct dinjonosis Hance the g A. CS Scanned with CamScanner

peco ICA ) pecs) o Chiapo ) + + OC pa-pa) + bo B C2,-€.) +$0 op " - opo thor = P202 + P²_2p 20 + ponor + 2 Pipo - 2 Pipoo + FM )o? -(ti?+ på + 2bit) 9 + 2b, po = 0 (p+poro CÞ,+ pojma + 2b1bo so Discriminant >0 4 4 7 0 (Pitpot - 4 (pit po ) x2 Pipo to (pr+ po 24 - { [+1+baº +(* =**)"} {4+bo) – 6-}} 20 (fitto) {(bitto) (Pr - po 2"} :-) > (0,+.) + JE.-bo) 7 2 (P. +},…) which is true 8 - 2 piko - +,+$,2 now theo If o=1 PCCBICA) = PCCB) provelance ire if the com of disease in sure event then the correct diagnosis the taro doehoors are independent. And this situation is boivial, apipo CB & CA are incependent iti o z by then also praspo CS Scanned with CamScanner
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