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# Probability Space - proof by induction watch

1. So far I've done the following:

Let then we get on the LHS while I get on the RHS which equals the LHS.

Then say it's true for hence I get

So then let and we get:

which turns into

and now I don't understand how I can turn the last 2 terms into
2. (Original post by RDKGames)

So far I've done the following:

[...]

which turns into

and now I don't understand how I can turn the last 2 terms into
What probability space axioms are you working with? One of the most fundamental is that if A and B are disjoint events (empty intersection) then . But in your proof, the need not be disjoint. So it looks like you need to do some complementing. Note that

EDIT: In your argument above, the indices i<j and i<j<k run only upto a, NOT upto a+1, so you actually need to show something more complicated than that the last two terms turn into (which I imagine is probably false in general).

A good idea is to note that is a union of events so you can use the inductive hypothesis again.
3. (Original post by RDKGames)
which turns into
I'd have to Google it myself, but I am wary of your working.

When you say "which turns into", what are the limits on the summation? E.g i<j. Is it , or . I suspect the former, though I could be misreading it.

Edit: I see theOldBean has now picked that up.

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