conditioning as restricting the sample space

Here is the most intuitive way to picture conditional probability. The whole sample space is the full set of possible outcomes, drawn as a region. When you learn that event B happened, you erase everything outside B — the outside is now impossible. B becomes your entire new universe. Conditioning is just zooming in on B and asking your old question inside this smaller world.

Concretely, take all the probability that used to live across the whole space and throw away the part outside B. The part inside B, namely P(A and B), is what survives — but it no longer adds up to 1, because you discarded the rest. So you rescale by dividing by P(B), the total surviving probability. That recovers a proper probability that sums to 1 over the new world B. This is exactly why P(A given B) = P(A and B) / P(B): the numerator is the surviving overlap, the denominator restores the total to 1.

This picture explains many surprises at a glance. 'At least one head' is not the same condition as 'the first coin is heads', so they shrink the space to different sub-worlds and give different answers. It also clarifies why order of information can matter, and it is the mental model behind the Monty Hall problem: when the host opens a door, he restricts the sample space in a way that is easy to get wrong if you do not track exactly which world you are now in.