Math Is Fun Forum

Ultraforce

Member

Registered: 2009-08-13

Posts: 1

Looks simple but I'm making a mistake somewhere...

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:

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)
= (1/2)p + P(choosing C)*P(B receives the letter from C)*P(I receive the letter from B)
= (1/2)p + (1/2)p*p
= (1/2)p + (1/2)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 (1/2)p + (1/2)p^2 - (1/4)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...

Last edited by Ultraforce (2009-08-14 00:00:57)

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Looks simple but I'm making a mistake somewhere...

Ultraforce, you need to understand the event first. Your event, if i get it correctly, is one but only one of A or C's letter reaches you. Since P{A or C}=P{A but not C}+P{C but not A} the calculation above is not right. when you add up probabilities you need to check if they have overlapping part first. if you add up two events like {A} and {C} you are counting {2 letters} twice. So an alternative would be P{A or C}=P{A}+P{C}-2P{2 letters} , and P{at least 1 letter}=P{A}+P{C}-P{2 letters}

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