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Can anyone explain this exercise to me?:

Quick exercise 2.5: Consider the sample space {a1 ,a2 ,a3 ,a4 ,a5 ,a6 } of some experiment, where outcome "a i" has probability "p i" for i =1,...,6. We perform this experiment twice in such a way that the associated probabilities are:

P((a i, a i)) = p i and P((a i, a j)) = 0 if i =/= j, for i, j = 1,...,6.

Check that P is a probability function on the sample space Ω = {a1,...,a6} × {a1,...,a6} of the combined experiment. What is the relationship between the first experiment and the second experiment that is determined by this probability function?

So the solution given is this:

2.5 Checking that P is a probability function Ω amounts to verifying that 0 ≤ P((a i,a j)) ≤ 1 for all i and j and noting that:

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The two experiments are totally coupled: one has outcome a i if and only if the other has outcome a i.

Please can someone explain this to me? :( What's happening here?

I know that all the probabilities in omega summed will equal to 1.. it has to, at least that's what I know. Given the P((ai, aj)).. aj will eventually be different than ai since we multiply the sample spaces and we will have 36 permutations... so this makes the probability... unequal, unfair... it's not like tossing a coin where you have 1/2 probability of an outcome, you should have in our case 1/36 to be equally likely, but given that if aj =/= ai then the P((ai,aj))=0... so there will be for sure one P that will be 0, UNLESS you set all the a's to be the same number.... so example would be W={a1 = 1, a2 = 1, a3 = 1, ....., a6 = 1}

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Please explain what you do not understand. –  Did Jan 2 at 20:44
    
I kinda understand the answer to the problem, but I don't understand his question... I would never think about producing the result that was given in the answer... I would be stuck.. I cant understand what's being asked of me to do. :( –  Chris Dobkowski Jan 2 at 20:51
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1 Answer

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You are making the assumption that the two experiments are independent, which they are not. Since the probability that the first experiment results in $a_1$ is $p_1$ and the chance that the two experiments both get $a_1$ is also $p_1$, the second experiment must result in $a_1$ every time the first one does. The same reasoning shows that the second experiment must have the same result as the first, whatever the result of the first.

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Still can't understand :( I know that ai = pi which means that the probability of ai happenign is pi, but where does aj come in? –  Chris Dobkowski Jan 2 at 21:02
    
ai is an event, pi is a probability. The point is that the chance of two ai's in succession is the same as the first ai. When they say $P(a_i,a_j)=0$ for $i \neq j$, you never have two different events in succession –  Ross Millikan Jan 2 at 21:31
    
Thank you very much!! :) –  Chris Dobkowski Jan 4 at 22:10
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