I am given positive integers $1,2,3,...,4k$.

Question

I am given a list of positive integers $1,2,3,...,4k$.

Let integer $n$ be the median of 3 randomly picked integers from the list.

What would be the exact probability that $k\leq n-1\leq 3k$?

Answer 1

The middle number of the three shall be the median. So, for the median $n$ to satisfy $k+1\leq n\leq 3k+1$, it means that at least one number is choosen from this partition of the set (let's call this subset $S$, $n<k+1$ as $A$ and $n>3k+1$ as $B$), and for that number to be the median we can have three cases:

Case 1: All three numbers are chosen from $S$

Case 2: Two numbers chosen from $S$, one from $A$

Case 3: Two numbers chosen from $S$, one from $B$

Case 4: One number is chosen from $S$, one from $A$, one from $B$

I believe the thinking part of the problem is done now, the rest should be relatively straightforward combinatorics