All of a sudden, I seem to be confused about the union of probabilities. I wanted to see how this would work out but ended up confusing myself even more. Here's a scenario I'm considering:(adsbygoogle = window.adsbygoogle || []).push({});

I have three friends. I know that if a friend posts me a letter upon being asked, it reaches me with a probability of p. I can contact my first friend A, directly but I would have to contact C through B. I wanted to calculate the probability of getting a letter from one of A or C. So, in the end, I have to get one letter atmost. To do this, I ended up saying the following:

P(getting at most one letter)

= P(I receive a letter from A or I receive a letter from C)

= P(I receive a letter from A) + P(I receive a letter from C)

= P(choosing A)*P(letter from A reaching me) + P(choosing C)*P(letter from C reaching me)

= [tex]\frac{1}{2}[/tex]p + P(choosing C)*P(B receives the letter from C)*P(I receive the letter from B)

= [tex]\frac{1}{2}[/tex]p + [tex]\frac{1}{2}[/tex]p*p

= [tex]\frac{1}{2}[/tex]p + [tex]\frac{1}{2}[/tex]p^{2}

I felt something was wrong so I ended up subtracting the P(getting a letter from both A and C) to make the solution as [tex]\frac{1}{2}[/tex]p + [tex]\frac{1}{2}[/tex]p^{2}- [tex]\frac{1}{4}[/tex]p^{3}from the axiom of probability of unions.

I've complicated the problem so much that I'm now confused... I am making a fundamental mistake in understanding the problem and hope someone can help me out in understanding it right...

PS: I made up this problem myself to try out something interesting so I might have been wrong in framing the question itself... If thats the case, please advice...

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# Looks like a simple probability calculation but am I getting it right?

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