Suppose there's items $A,B,C,D,E$ where the desired pick will be to pick $C$. At first try, I randomly picked $B$, hence now the pick-pool will be $A,C,D,E$ and second try I picked $E$ and now it's $A,C,D$ and so on.
So if there's $n$-items and $x$-tries ($x \leq n$). What is the probability that I get the desired item out of these random items after $x$-tries. If we have $x=1$, then the probability is $1/n$ and if $x=n$, then it's $1$
On
Thought experiment.
You have $n$ items, one of them with the label X.
Randomly assign each of the items a number from $\{1,2,\cdots,n\},$ without replacement.
The assigned number represents on which turn the item will be picked.
Given any two numbers $i,j$ such that $i,j \in \{1,2,\cdots, n\}$ and $i \neq j$, is the item labeled X more likely to be assigned the number $i$ rather than the number $j$?
Is the item labeled X more likely to be assigned the number $j$ rather than the number $i$?
This is just the probability that if you pick $x$ items out of $n$, the desired object will be among those you picked, so the probability of success is $$\frac xn$$