Let $x_n=\{1,2,3,...,n\}$ and let a subset $A$ of $X_n$ be chosen so that every pair of elements of $A$ differ by at least $3$. When $n=10$, let probability that $1\in A=p$ and probability that $2\in A=q$. Find $p$ and $q$.
I am not being able to find either the sample space or decide how to find the number of events in $p$ and $q$.
How should I do it? Thanks in advance!
Lulu's hint is good and his formula is correct, but argument is not. Say subset is good if every two differ for at least $3$. So, if $s_n$ is a number of good subsets in $\{1,2,...,n\}$ then we have $$s_n = s_{n-3}+s_{n-1}$$
(that is, if $n$ is in a good subset then we must take the rest of elements in $\{1,2,...,n-3\}$ else we take elements from a set $\{1,2,...,n-1\}$).
Now let $a_n$ be a number of good subsets with $1$ in it. Then $a_n=s_{n-3}$
Since first $11$ terms is (starting with $s_0$) $$1,2,3,4,6,9,10,15,19,25,34$$ we have
$$p = {a_{10}\over s_{10}}= {s_{7}\over s_{10}} = {15\over 34}$$
You can procede similary for $q$ (note that $b_n=a_{n-4}$)