I have a homework problem that I got wrong, and I'm trying to understand why. It's an application of Bayes Rule - we are trying to determine the posterior probability of I, after three tests - 2 positive, and 1 negative.
If I want to know P(I|+,+,-), I understand that
P(I|+,+,-) = P(+,+,-|I)*P(I)/P(+,+,-)
When I tried to figure this out, I changed this to
P(+|I)*P(+|I)*P(-|I)*P(I)/(P(+|I)*P(+|I)*P(-|I)*P(I)+P(+|_I)*P(+|_I)*P(-|_I)*P(_I))
But that was incorrect. For the correct solution, they just figured out P(+), which was P(+|I)*P(I)+P(+|_I)*P(_I), and used P(+)*P(+)*P(-)

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