Thread: Disease prevalence

I'm stumped on this question.

Conditions:

Let PPVa = P[D|Ta]
Let PPVb = P[D|Tb intersection Tb]

Let SigmaA = P[Ta|D] = P[Ta|D intersection Tb]
Let KappaA = P[Ta complement|D complement]

Let SigmaB = P[Tb|D]
Let KappaB = P[Tb complement|D complement]

P[D|Ta intersection Tb] > P[D|Ta]

Attempt at a solution:

P[D intersection Ta intersection Tb]/P[Ta intersection Tb] > P[D intersection Ta]/P[Ta] (by condtional probability)

P[D]P[Tb|D]P[Ta|D intersection Tb]/P[Ta intersection Tb] > P[D]P[Ta|D]/P[Ta] (by event composition)

P[Tb|D]/P[Ta intersection Tb] > 1/P[Ta]

P[Tb|D] > P[Ta intersection Tb]/P[Ta]

SigmaB > P[Tb|Ta] (by conditional probability)

I don't know where to go from there. Some help would be greatly appreciated. The answer is supposed to be: SigmaB > 1 - KappaB.