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

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

## Homework Statement

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".

## 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).

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## Answers and Replies

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Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

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".

## 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 $M_i \backslash (M_j \cup M_k) \forall i$.

-------------

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).
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

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

So what I've done is correct?

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 :).

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Ray Vickson
Homework Helper
Dearly Missed

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 :).
It is obviously true.

RGV

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