Consider a situation where someone is diagnosed as positive for an illness with the following characteristics:
$$P(\text{True(+))}=0.005$$ $$P(\text{False(+))} = 0.05$$
Consider being diagnosed with the same result for a second time. The diagnoses are independent, so you would square the above for the new probabilities of True(+) and False(+). (I think?)
So this gives: $$P(\text{True(+))}=0.000025$$ $$P(\text{False(+))} = 0.0025$$
I don't see how this makes realistic sense. Surely if you get diagnosed with the same illness more than once, then the probability of it being correct cannot decrease. Imagine getting diagnosed 100 times, the above would suggest that now there is a lower chance of having the illness than when you were first diagnosed.
Where is my thinking going wrong? Please just tell me what to think about/consider so that I can arrive there myself. (I can post a full solution later on.)