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Based on https://en.wikipedia.org/wiki/Law_of_total_probability, the LOTP for the discrete case can be written as a summation over the joint probability of the event we're computing the probability for and all possible values for some discrete variable, e.g., Wikipedia has

$$ P(A) = \sum_{n}P(A, B_n) $$

For the continuous case, Wiki shows $$ P(A) = \int P(A | X = x)f_X(x)dx $$

where $X$ is a continuous random variable. The distinction here is that a conditional probability is used instead of a joint probability like in the discrete case. Is that because you can't really write something like $$ P(A) = \int P(A, X = x) dx $$

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The problem I see with this expression is $P(A, X = x) = 0$ since $P(X = x) = 0$. Is this the reason why the conditional probability is used when using LOTP and conditioning on a continuous random variable?

Can you alternatively write:

$$ P(A) = \int f(A, X = x) dx $$

where $f$ is now a joint pdf of the event $A$ and continuous random variable $X$?

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$A$ isn't a random variable, so it doesn't have a joint distribution (what values would it take?). And so only the first is correct. However, if we are talking about random variables, you can obtain the marginal distribution of any discrete/continuous random variable by marginalizing a discrete/continuous random variable out from the joint distribution.

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