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mathmari

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MHB

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The percentage of people that have a disease A is $0,01$.

We apply twice a test for that disease, each of which give the correct answer with probability $0,95$.

What is the probability that someone has that disease if at least one test is positive and what is the probability if both tests are positive? I have done the following:

We have that $$P(\text{at least one positive})=1-P(\text{no positive})=1-P(NN)$$ where N: "negative". Let D be the event "has the disease".

Then we have that $$P(NN)=P(NN\cap D)+P(NN\cap D^C)=P(NN\mid D)\cdot P(D)+P(NN\mid D^C)\cdot P(D^C)$$ We have that $P(D)=0,01$ and $P(D^C)=0,99$.

Does it hold that $P(NN\mid D)=P(N\mid D)^2$ and $P(NN\mid D^C)=P(N\mid D^C)^2$ ?

If yes, does it hold that $P(N\mid D)=0,05$ and $P(N\mid D^C)=0,95$ ?

That would mean that we get $$P(NN)=0,05^2\cdot 001+0,95^2\cdot 0,99$$ Is this correct? (Wondering)