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January 20th, 2015, 12:51 PM   #1
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Bayes' Theorem

Hi,

I've gotten a hang of conditional probability and mutually exclusive events but I always get stuck when trying to understand Bayes' theorem. I've read a couple good examples about false positives and false negatives during screening for example but still can't wrap my head around the formula :

I don't understand how we jump from

$\displaystyle P(A intersection B_j)=P(B_j intersection A)=P(B_j)P(A|B_j)$

to

$\displaystyle P(B_j|A)=(P(B_j)P(A|B_j))/(P(A))$

Why are we dividing by P(A) ?

Thanks,
amphi

Taken from here: Bayes' Theorem -- from Wolfram MathWorld
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January 20th, 2015, 01:19 PM   #2
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It's a small piece of algebraic manipulation.

$P(B_j|A) = \dfrac{P(B_j\cap A)}{P(A)}$

By the formula for conditional probability. From your given formula, we know that

$P(B_j\cap A) = P(B_j)P(A|B_j)$

Substituting this into formula for $P(B_j|A)$ gives

$P(B_j|A) = \dfrac{P(B_j)P(A|B_j)}{P(A)}$

As required.
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January 20th, 2015, 11:43 PM   #3
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Thanks !
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