Calculating the probability of one event based on other events

camel

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Jan 19, 2018
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Hi all,

I came across this probability problem:


Let A1, A2, ..., An be events. Let Bk be the event that at least k of the Ai occur, and Ck be the event that exactly k of the Ai occur, for 0<=k<=n. Find a simple expression for P(Bk) in terms of P(Ck) and P(Ck+1).



My initial thought is that since we need to express P(Bk) in terms of P(Ck) and P(Ck+1), there must be some way to expression Bk in terms of Ck and Ck+1, but it seems P(Bk) should be equals to P(Ck) + P(Ck+1) + P(Ck+2) + ... + P(Cn), instead of just Ck and Ck+1, so I am stuck here, could any body provide some hint?

Thanks.
 
Let A1, A2, ..., An be events. Let Bk be the event that at least k of the Ai occur, and Ck be the event that exactly k of the Ai occur, for 0<=k<=n. Find a simple expression for P(Bk) in terms of P(Ck) and P(Ck+1).



My initial thought is that since we need to express P(Bk) in terms of P(Ck) and P(Ck+1), there must be some way to expression Bk in terms of Ck and Ck+1, but it seems P(Bk) should be equals to P(Ck) + P(Ck+1) + P(Ck+2) + ... + P(Cn), instead of just Ck and Ck+1, so I am stuck here, could any body provide some hint?

I'm wondering if the problem was stated (or copied) incorrectly, because it is easy to express P(Ck) in terms of P(Bk) and P(Bk+1).
 
I'm wondering if the problem was stated (or copied) incorrectly, because it is easy to express P(Ck) in terms of P(Bk) and P(Bk+1).

Yes, I think your comment makes sense. It is more logically smooth to understand in your way.
 
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