Probability question - step function?

Question

How would you figure out the probability function of this scenario:

There are 9 pieces of paper with number 1-9 (one each) in a box. Select the paper 9 times (with replacement) then select a positive number x (can be anything). What is the probability that x is higher than the 3rd highest of 9 numbers selected.

I have some idea that the final probability function will look like a step function. It looks quite simple when I tried to characterize each combination. For example,

with probability $(1/9)^9$, all selected numbers are 1, then Pr(Win) = 1 if x > 1, = 3/10 if x = 1, = 0 if x < 1
with probability $(1/9)^9$, all but one selected numbers are 1, the rest is 2 then Pr(Win) = 1 if x > 1, = 3/10 if x = 1, = 0 if x < 1

and so on.

However, I am lost when trying to characterize them together (i.e., combining all those combination). Any comments, suggestions, hints, are greatly appreciated.

Thank you,

$x$ can be any positive number??? In that case, the probability of it being larger than the 3rd highest is $100%$. — barak manos, May 28 '16 at 16:36
Why? if x is 1 and all selected numbers are 9 then x is smaller than the 3rd highest number isn't it? — gdsquare, May 28 '16 at 16:55
Yes, but there are infinitely many values of $x>9$ and only finitely many values of $x\in[1,9]$, so the probability of choosing the former vs. the latter is $100%$. — barak manos, May 28 '16 at 16:57
@barakmanos: There is no uniform distribution over $\mathbb N$. — joriki, May 28 '16 at 22:49

Probability question - step function?

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