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Let $S$ be the set of natural numbers from zero to $n$, namely $S = \{ s : s∈\mathbb{N} ∧ s ≤ n \}$.

With each turn we pick a random number from the set. If we get one specific number (let's say $0$), the experiment is complete. If we get any other number, we go on with the next turn (hoping to get $0$ and going on with yet another turn otherwise).

Is it possible to prove that such an experiment always ends with a finite number of turns?

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The set $\{0,1,2,\ldots,n\}$ has $n+1$ members.

Getting a number other than $0$ on the first trial has probability $n/(n+1).$

Getting a number other than $0$ on all of the first $N$ (not $n$) trials has probability $\left( \dfrac n{n+1} \right)^N.$

Getting a number other than $0$ on all trials in an infinite sequence of trials therefore has probability${}\le\left( \dfrac n{n+1}\right)^N.$

The only nonnegative number that is${}\le\left(\dfrac n {n+1}\right)^N$ for all $N\in\{1,2,3,\ldots\}$ is $0.$

So the probability of getting an infinite sequence of nonzero numbers is $0.$

However, the set of all infinite sequences of numbers in the set $\{1,2,3,\ldots,n\}$ is not empty. This is a nonempty subset of the probability space, but its probability is $0.$

It is analogous to throwing a dart at a square and asking for the probability that it lands exactly on the diagonal. The area of the diagonal is $0$ but the diagonal is not empty.

For many purposes in probability and statistics, sets of probability $0$ can be neglected, just as if they are not there.

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The probability that the game is still going after $k$ turns is $\left(\frac{n-1}{n}\right)^k$, because you picked a number other than your target number (0 in your example) $k$ times in a row. Let the random variable $X$ denote the number of turns needed to finish the game. You can see that: $P(X=k) = \left(\frac{n-1}{n}\right)^{k-1} \frac{1}{n}$ for $k>0$.

Now we can compute $P(X < \infty) = \frac{1}{n}\sum_{k=0}^{\infty} \left(1 - \frac{1}{n}\right)^k = \frac{1}{n}\times \frac{1}{1 - \left(1 - \frac{1}{n}\right)} = 1$.

Thus, the number of turns required for the game to finish is almost surely finite.

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