Conditional probability
Once you learn that one event has happened, how does the chance of another one change?
Where you are 3 inside
What this area is about
New information changes a chance. If you know an event A has happened, the outcomes outside A are ruled out, and the chance of B has to be measured inside A alone. That new number is the conditional probability P(B \mid A), defined only when P(A) > 0. This area computes it, uses it to build an overall chance from separate cases, and turns it around.
How these fit together
The definition. Finding a conditional probability divides the chance that both happen by the chance of the event conditioned on: P(B \mid A) = P(A \text{ and } B) / P(A), for P(A) > 0. It also tests whether knowing A changes the chance of B at all.
A result. The law of total probability splits the outcomes into cases that do not overlap and together cover everything, then adds each case’s share of the event.
A procedure. Bayes’ rule reverses the condition: from the chance of the evidence within each case to the chance of a case given the evidence.
A path through it
Take them in this order: the formula first, then the law of total probability, then Bayes’ rule, which divides one path of a tree by the total that the law of total probability supplies.
Finish here and you can later see what a p-value does and does not say: it is a chance computed on the condition that the null hypothesis is true.