Question

State and prove Baye's theorem.

Answer 1

Bayes theorem is stated as $P\left(A/B\right)=\frac{P\left(B/A\right)P\left(A\right)}{P\left(B\right)}$

Proof$:$

We can do it from set theory applied to conditional probability

$P\left(A/B\right)=\frac{P(A\cap B)}{P\left(B\right)}$

Likewise $P\left(B/A\right)=\frac{P(A\cap B)}{P\left(A\right)}$

Rearrange to get common terms as follows , $P(A\cap B)=P\left(A/B\right)P\left(B\right)=P\left(B/A\right)P\left(A\right)$

Therefore we get $P\left(A/B\right)=\frac{P\left(B/A\right)P\left(A\right)}{P\left(B\right)}$

This is basic statement of Bayes theorem