# Thread: Compuite P{X < Y} conditioned on the value of Y; intuition behind it.

1. ## Compuite P{X < Y} conditioned on the value of Y; intuition behind it.

Dear people,

I have a question concerning the following steps to compute P{X < Y}.
The steps shown are the following:

P{X < Y} =

I do not completely get the intuition behind this.
First you condition on Y = y, then you change the Y in y in the next step and then you remove the conditioning on Y = y because that obviously does not give you anymore information now that P(X < y). But couldn't you use P(X < y) right away?
If someone could give some additional help that would be nice.

Thank you!

2. ## Re: Compuite P{X < Y} conditioned on the value of Y; intuition behind it.

I am not sure what exactly giving you trouble. I try to explain in the following way:

The first step here is necessary - and is more general. I believe that in this question it was assumed that are independent so that the condition probability can be safely changed to the ordinary one in the 3rd line. But in general they can be dependent and you cannot do so.

Or you may view this in other perspective:

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