Question

State and prove Bayes' theorem

Answer 1

The probability of two events $A$ and $B$ happening, $P(A\cap B)$, is the probability of $A$ $\left(P\right(A\left)\right)$,times the probability of $B$ given that $A$ has occurred $\left(P\right(B|A\left)\right)$.

$P(A\cap B)=P\left(A\right)P\left(B|A\right)$ ....$\left(1\right)$

On the other hand, the probability of $A$ and $B$ is also equal to the probability of $B$ times the probability of $A$ given $B$.

$P(A\cap B)=P\left(B\right)P\left(A|B\right)$ ....$\left(2\right)$

Equating the two yields:

$P\left(B\right)P\left(A|B\right)=P\left(A\right)P\left(B|A\right)$ .....$\left(3\right)$

and thus

$P\left(A|B\right)=P\left(A\right)\frac{P\left(B|A\right)}{P\left(B\right)}$ ....$\left(4\right)$

This equation, known as Bayes' theorem is the basis of statistical inference.