A single die roll shows why joint probabilities are not simply multiplied
Roll one ordinary die and ask two questions at once: is the result even, and is it prime? Only the 2 answers yes to both, so that combination has a one-in-six chance. A joint probability distribution is the full table of such combined chances, and it holds information that the separate odds alone cannot give.
Whenever two or more random quantities are defined on the same experiment, their joint distribution gives the probability of every combination of values. With two variables it is called bivariate, but the idea extends to any number. For the die, the four combinations work out neatly. Neither even nor prime covers only the 1, at one sixth. Even but not prime covers 4 and 6, at two sixths. Prime but not even covers 3 and 5, also two sixths. Both applies to the 2 alone, at one sixth. As with any distribution, the four cells add up to exactly 1.
From such a table you can recover each variable's own odds, called its marginal distribution, by adding across rows or down columns while ignoring the other variable. You can also read off conditional distributions: the chances for some variables once others are known. For continuous quantities the table becomes a density, and the adding is replaced by integration.
The simple rule of multiplying separate probabilities works only when the variables are independent. Two urns, each holding twice as many red balls as blue, give a clean example: draws from different urns do not influence each other, so the chance of red then red is two thirds times two thirds. Two fair coin flips behave the same way, each of the four outcomes carrying a quarter. The die questions are different because both describe one roll, so the joint table must be built by counting outcomes directly.
More generally, a joint probability can always be written as a chain of conditional steps: the chance of the first value, times the chance of the second given the first, and so on. That identity is known as the chain rule of probability, and it underlies much of modern statistical modelling.
Source: Joint probability distribution
