bayes formula

let $\text{[math]}$ be a countable set and let $\text{[math]}$ be pairwise disjoint sets with $\text{[math]}$ . then, for any $\text{[math]}$ with $\text{[math]}$ and any $\text{[math]}$ ,
$\text{[math]}$ [cite:;taken from @klenke_prob_2020 theorem 8.7]

let $\text{[math]}$ be a broken link: blk:def-partition of the sample space, and let $\text{[math]}$ be any set. then, for each $\text{[math]}$ ,
$\text{[math]}$ [cite:;taken from @berger_inference_2002 theorem 1.3.5]

in simpler terms,

the probability of A occuring, knowing that B has occured is
$\text{[math]}$ two events are independent if knowing one event occurred doesn't change the probability of the other event. otherwise they are both dependent on one another

e. two events, A and B, are independent if:

$\text{[math]}$