Given the problem of a patient taking a test for a disease where having the disease is denoted by X and the a positive test is denoted by Y, - the rate of occurrence of the disease in the general population is 1%, - the probability of a false positive is 3%, - the odds of getting tested positive is 90% if you have the disease

Bergsteinj0 2022-09-08 Answered
Given the problem of a patient taking a test for a disease where having the disease is denoted by X and the a positive test is denoted by Y,
- the rate of occurrence of the disease in the general population is 1%
- the probability of a false positive is 3%
- the odds of getting tested positive is 90% if you have the disease
is it appropriate to solve the following through rearranging the total probability
P ( A ) = P ( A | B ) P ( B ) + P ( A | N o t B ) P ( N o t B )
Into
P ( A | B ) = ( P ( A ) P ( A | B ) P ( B ) ) / P ( N o t B )
and then getting P(B) from the total probability
P ( B ) = P ( B | A ) P ( A ) + P ( B | N o t A ) P ( N o t A )
then given B is binary P(Not B) from
1 = P ( B ) + P ( N o t B )
and getting P(A|B) from Bayes
P ( A | B ) == P ( A ) P ( B | A ) / P ( B )
and then substituting that all back into the first equation to get the result
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Answers (1)

Quinn Alvarez
Answered 2022-09-09 Author has 13 answers
We have by Bayes' rule
P ( X | Y c ) = P ( Y c | X ) P ( X ) P ( Y c | X ) P ( X ) + P ( Y c | X c ) P ( X c ) .
Your setup has P ( X ) = 0.01 , P ( Y | X ) = 0.9 , P ( Y | X c ) = 0.03.
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