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

[operationsres]

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

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

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).

 

[Ray Vickson]

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

 

[operationsres]

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

So what I've done is correct?

 

[operationsres]

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 :).

 

[Ray Vickson]

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

 

[operationsres]

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