Conditional and Joint Probability

Conditional probability answers a narrower question than it first sounds like. 'Given that someone owns a dog, what's the chance they own a leash?' doesn't ask about everyone surveyed. It throws out everyone who doesn't own a dog and asks only about the dog owners left behind. That's the whole trick: conditioning on an event shrinks your sample space down to just the people or outcomes satisfying that event, then asks what fraction of that smaller group also satisfies the second event.

In a two-way table, this shows up as picking one row (or one column) and working only within it. Joint probability, by contrast, uses the whole table as its denominator: it's the count in one specific cell divided by the grand total of everyone surveyed. Conditional probability uses only the row or column total as its denominator. Same numerator cell, different bottom number. That's also why P(A given B) and P(B given A) are usually different: both divide the same joint cell, but one divides by B's total and the other divides by A's total. They only match when A and B take up the same share of the whole group.

This sits right next to independence: two events are independent exactly when conditioning changes nothing, so P(A given B) equals plain P(A). It also sets up expected value and simulation questions, which often start by asking you to find a conditional probability first, then use it as a weight.