Difference Between joint probability distribution and conditional probability distribution?

Broadly speaking, joint probability is the probability of two things* happening together: e.g., the probability that I wash my car, and it rains. Conditional probability is the probability of one thing happening, given that the other thing happens: e.g., the probability that, given that I wash my car, it rains.

Consider the space of all four combinations:

• I wash my car, and it rains.
• I wash my car, and it doesn't rain.
• I don't wash my car, and it rains.
• I don't wash my car, and it doesn't rain.

Quantitatively, the difference is that in the case of joint probability, we evaluate across the space of all four combinations: The joint probability is the the first, divided by the sum of all four. (Note: If we're talking about a probability space, the sum of all four is everything—i.e., it has probability one.)

In the case of conditional probability, we only examine those cases where I wash my car (the antecedent). So the conditional probability is the first, divided by the sum of the first two.

*More than two things could be involved. But the basic idea can most clearly be illustrated with two things.