Originally Posted by

**Moo** Hello,

So you're looking for P(D|T)

$\displaystyle \mathbb{P}(D|T)=\frac{\mathbb{P}(D \cap T)}{\mathbb{P}(T)}$

But we know that $\displaystyle \mathbb{P}(D \cap T)=\mathbb{P}(T|D)\mathbb{P}(D)$

So $\displaystyle \mathbb{P}(D|T)=\frac{\mathbb{P}(T|D)\mathbb{P}(D) }{\mathbb{P}(T)}$

Now what is $\displaystyle \mathbb{P}(T)$ ?

$\displaystyle T=(T \cap D) \cup (T \cap D^c)$ (because D and its complementary $\displaystyle D^c$ are disjoint)

And we also have $\displaystyle (T \cap D) \cap (T \cap D^c)= \emptyset$

Hence $\displaystyle \mathbb{P}(T)=\mathbb{P}(T \cap D)+\mathbb{P}(T \cap D^c)$

$\displaystyle =\mathbb{P}(T|D)P(D)+\mathbb{P}(T|D^c)P(D^c)$

Therefore, $\displaystyle \boxed{\mathbb{P}(D|T)=\frac{\mathbb{P}(T|D) \mathbb{P}(D)}{\mathbb{P}(T|D)P(D)+\mathbb{P}(T|D^ c)P(D^c)}}

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