# Eli5 The sum rule for probability.

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The sum rule: the probability of an event is the sum of all the joint probabilities with another event. P(X=x) = (y ∃ Ω) Σ P(X=x,Y=y)

Is this the same as the Law of Total Probability?

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First, you should be careful with your terms. “Event” is not interchangable with “random variable”, and your statement is one about random variables: events either happen or don’t, random variables take on numerical values. An event is a *set* of values for your random variables.

That said, there’s two ways to approach this.

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One way is to think in terms of the *joint* probability distribution – that is, think about every possible *pair* of values X and Y could take on, and ask what the probability of that *pair* of values is. For example, if X and Y are both fair dice, there are 36 ordered pairs starting with (1,1), (1,2), (1,3),… and ending with …(6,4), (6,5), (6,6). Each of those 36 pairs has some probability – in the case of fair dice, they happen to all be equal – and those probabilities are a complete description of the entire system.

But you’re often interested in a *marginal* probability – that is, you’re interested in the distribution X takes on, without reference to Y. That is, we want to take all of those ordered pairs and group them by the X values, or a bit more formally, we want to say P(X = x) = P(any of the pairs for which X = x and Y = anything)

Since only one of the ordered pairs can happen, each ordered pair is mutually exclusive with any or all of the others – if any one of them happens, the others do not, and vice versa. And in general, the probability of any one of several mutually-exclusive events occurring is just the sum of each of their individual probabilities.

So P(X = x), our marginal, equals P(any of the pairs with X = x and Y = anything) by definition, and this in turn equals sum(P(X = x and Y = y)) for each individual y because we’re adding up all the mutually exclusive probabilities of each of the pairs.

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Another way to think about it is, yes, that it’s just the law of total probability.

Define the event A to be the set of all ordered pairs where X = x. Then by the law of total probability:

P(X = x) = P(A) = sum(P(A intersect B_y)) for any set of pairwise disjoint events B_y. Then define your set of disjoint events B_y to just be the sets of ordered pairs where Y = y. Intuitively, you can think of the event A as identifying a particular “row” of a table of ordered pairs, while the events B_y identify “columns”. Then:

P(X = x) = P(A) = sum(P(A intersect B_y)) = sum(P(X = x, Y = y)).

and you’re done. (The continuous case requires a little more formality, but it works out the same way.)

Intuitively, what you’re saying is “the total value in a row is the sum of all the individual values across that row”. Which is very straightforward.