Conditional Probability

Someone rolls a fair die behind a screen and tells you one thing: the result is even. What is the chance it is a 6 now? Not \(1/6\) any more. The words the result is even quietly threw away three of the six faces. Only \(\{2,4,6\}\) are still possible, so a 6 is one of three equally likely outcomes — a chance of \(1/3\).

That is the entire idea of a conditional probability. Writing \(P(A\mid B)\) — read the probability of A given B — means: assume B has happened, so B is now the only world that exists, and ask what share of that world is also in A.

Dividing by \(P(B)\) is a re-basing step. Before you knew anything, every probability was measured against the whole sample space, which sums to 1. Once you condition on B, the leftover B-world is smaller than 1, so you rescale it back up to a full 100% by dividing by its size \(P(B)\). After that division the conditional probabilities of everything inside B add up to 1 again — a proper probability world of its own.

The numerator \(P(A \cap B)\) has to be the overlap, not all of A. Parts of A that sit outside B are gone: they were ruled out the moment B was announced.

The simulation below shows 100 equally likely outcomes as a 10 by 10 grid, so a probability is simply a count out of 100. Use the sliders to set how many outcomes fall in event A (purple), in event B (blue), and in the overlap that is in both (green).

Then press Condition on B. Every square outside B fades out, and the readout's denominator drops from 100 to just the number of B cells. In that greyed, smaller world, \(P(A\mid B)\) is nothing more than the share of what remains that is green — the outcomes that are in A as well.

Most of the time you meet conditional probability as a table of counts. Picture 200 commuters, sorted two ways — how they travelled and whether they arrived on time:

• Train riders: 120 — of these, 90 were on time and 30 were late.
• Drivers: 80 — of these, 50 were on time and 30 were late.

The column totals are 140 on time and 60 late, adding to 200. To read a conditional like \(P(\text{Late}\mid\text{Drove})\), keep only the row you are conditioning on — the 80 drivers become your new denominator — and count the late ones inside it: \(30/80 = 0.375\). Compare that with the overall \(P(\text{Late}) = 60/200 = 0.30\): driving raises the chance of being late, so these two events are not independent.