# Find expression for event exactly one of A,B,C occurs

1. Aug 8, 2012

### operationsres

Find expression for event "exactly one of A,B,C occurs"

1. The problem statement, all variables and given/known data

Let $A,B,C \in F$ be three arbitrary events. F is a σ-algebra. Find an expression for the event "exactly one of A,B,C occurs".

3. The attempt at a solution

Define $i,j,k$ s.t. $i,j,k \in \{1,2,3\} \wedge i \not= j \not=k$. Also, define $M_1 := A, M_2 := B, M_3 := C$.

Then the event in question is $M_i \backslash (M_j \cup M_k) \forall i$.

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Is this correct? I've never seen this "notation" used in my lecture slides so I'm not sure (I'm trying to get out of writing a silly amount of unions and then simplifying).

Last edited: Aug 8, 2012
2. Aug 8, 2012

### Ray Vickson

Re: Find expression for event "exactly one of A,B,C occurs"

Yes, $A\text{ only} = A \cap (B \cup C)^c,$ etc, where $D^c$ denotes the complement of a set D. As to the probability that exactly one occurs, see Feller, "Introduction to Probability Theory and its Applications", Vol I, (Wiley), which gives expressions for P{exactly k events occur} for k = 0, 1, 2, ... among n events. You can also find similar developments in the various Probability books by Sheldon Ross.

RGV

3. Aug 8, 2012

### operationsres

Re: Find expression for event "exactly one of A,B,C occurs"

Thanks, I'll borrow that book from my uni's library.

So what I've done is correct?

4. Aug 8, 2012

### operationsres

Re: Find expression for event "exactly one of A,B,C occurs"

Actually I believe that I've made a mistake in my OP.

The correct notation for the event would be $\bigcup_{i=1}^3 M_i \backslash (M_j \cup M_k)$ for $i \not= j \not= k$ and $j,k \in \{1,2,3\}$.

Still would like confirmation that this is correct :).

Last edited: Aug 8, 2012
5. Aug 8, 2012

### Ray Vickson

Re: Find expression for event "exactly one of A,B,C occurs"

It is obviously true.

RGV

6. Aug 8, 2012

### operationsres

Re: Find expression for event "exactly one of A,B,C occurs"

Thanks for your help. Just wasn't sure of the notation :)