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: 


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

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

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

*$A_2^c ∩ A_3$
My attempt: 


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*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. 

We are dealing with $A_{1}\subsetneq A_{2}\subsetneq A_{3}\subsetneq A_{4}\subsetneq A_{5}$
so that: 


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

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

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

*$A_{2}^{\complement}\cap A_{3}$
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 $2$.
