Possible outcome for an infinite number of rolls of a die

#1
Suppose the following:
I have a die with 10 faces.
9 faces say 'lose' and 1 face says 'win'.
I repeatedly roll the die forever (i.e. an infinite number of rolls).

Is it guaranteed that at some point between now and infinity, the number of times the die has landed on the 'win' face, will be greater than the number of times it has landed on 'lose'?



Here are my thoughts so far:

I know that as the number of rolls increases, the ratio of 'loss' to 'win' should converge to 0.9.

However, I reason that at any given instant between now and infinity, there is a certain probability that a sufficiently large string of 'win's will occur to result in the overall number of 'wins' being greater than the overall number of losses.

However, I also note that this probability tends to 1/infinity (i.e. “1 over infinity”) as the total number of rolls increases.

I wonder can this problem be treated like an infinite series with a higher power in the denominator, such that there is no guarantee that the win rate will ever be greater?

Or does the simple fact that we are dealing with probabilities mean that it’s always possible, and therefore, it must happen sometime between now and infinity?
 

BGM

TS Contributor
#2
Let \( A_n \) be the event that there is more "win" than "lose" in the first \( 2n - 1 \) rolls for the first time, \( n = 1, 2, 3, \ldots \)

and you want to know something like \( \Pr\left(\bigcup_{n=1}^{\infty}A_n\right) \)

Actually if you treat it as an asymmetric random walk (with the position being the net difference between "win" and "lose"), then actually it is a transient random walk - the probability that it will ever return to 0 (equal numbers of "win" and "lose") is less than 1.

Therefore the above interested quantity is non-zero but less than one. i.e. It is not almost surely happen.