Probability - Describing mathematical events in words Consider a target that consists of $5$ concentric circles with radii  $1, 2, 3, 4, 5$ respectively. An event $A_j$, $j \in \{1,..,5\}$  describes a shot landing in the circle with the radius $j$. Describe the following events in words: 


*

*$\bigcap\limits_{k=1}^{3} A_{k}$

*$\bigcup\limits_{k=1}^{4} A_{k}$

*$ (A_3 \setminus A_2) ∪ (A_2 \setminus A_3)$

*$A_2^c ∩ A_3$
My attempt: 


*

*The shot lands in the circle with radius $1$ (Question: Since we're looking at the intersection of the sets, what role does $3$ play here? Wouldn't we be describing the same event if we let $k$ go to $4$ and $5$?)  

*The shot lands in circle $1, 2, 3$ or $ 4. $

*The shot lands in circle $2$ or $3$. 

*The shot lands in circle $3$. 
Are my answers correct? Any help would be much appreciated. 
 A: First of all, the definition of the events $A_j$ requires a little bit of clarification. I read it as $A_j$ meaning that the shot lands anywhere inside the corresponding circle. From this point of view, if you hit the center of the target, this outcome simultaneously belongs to all $A_1$, $A_2$, $A_3$, $A_4$, and $A_5$. And my answers below are based on this reading of the question. An alternative interpretation could be that each $A_j$ means being the ring between two concentric circles (but this seems unlikely to me).


*

*Your answer is correct. And yes, you're absolutely right — $3$ isn't really significant here, as there will be the same answer if we let $k$ go up to $4$ or $5$. But it's a textbook exercise, and they had to ask something specific, so for example but with no special significance they put a $3$ there. :-)

*Well, this is true, but this is not the correct final answer, because it can be simplified. Since the circles are concentric, being anywhere in circle $1$ or $2$ or $3$ or $4$ is equivalent to the being anywhere in which circle?

*No, this isn't correct. Note that $A_2\setminus A_3=\varnothing$, since $A_2$ lies entirely inside $A_3$.

*Not correct either. Think about what $A_2^c$ represents.
A: Your answer to 1 is correct.  If you added in the other two circles it wouldn't change anything because $A_1$ is a subset of all the others.  Any intersection that includes $A_1$ will just be $A_1$ in this case.  
Your answer to $2$ is correct but may be marked down.  As in 1, you are expected to realize that $A_4$ is a superset of the other three, so the union is just $A_4$.  
For 3 it must land in circle $3$ but not in circle $2$.  That is the point of the setminus.  
For 4 it is the same answer as $3$.
A: We are dealing with $A_{1}\subsetneq A_{2}\subsetneq A_{3}\subsetneq A_{4}\subsetneq A_{5}$
so that: 


*

*$\bigcap_{k=1}^{3}A_{k}=A_{1}$

*$\bigcup_{k=1}^{4}A_{k}=A_{4}$

*$\left(A_{3}\setminus A_{2}\right)\cup\left(A_{2}\setminus A_{3}\right)=\left(A_{3}\setminus A_{2}\right)\cup\varnothing=A_{3}\setminus A_{2}$

*$A_{2}^{\complement}\cap A_{3}$
In words:


*

*shot ends in circle with radius $1$.

*shot ends in circle with radius $4$ (which does not exclude that it ends in circles with smaller radius).

*shot ends in circle with radius $3$ and not in circle with radius $2$.

*shot ends in circle with radius $3$ and not in circle with radius $2$.
