Is ultimately rolling a 6 on a dice guaranteed?

Question

I roll a dice repeatedly and stop only when I roll a 6. Can I roll infinitely without stopping?

Here's where I am: rolling infinitely without rolling a 6 contradicts the fact that rolling a 6 has nonzero probability, which is a contradiction. Therefore one cannot.

Is this reasoning correct, and would it be possible to base an important proof regarding whether a certain process stops, upon this theorem?

Answer 1

A further point: "has probability zero" and "cannot happen" are not the same thing. All impossible events have probability zero, but there are possible events that have probability zero as well. These tend to turn up when you have infinitely many outcomes. For example, if you choose a number in $(0,1)$ uniformity at random, the probability that it is $0.4858282$ is zero. However, it is possible for that to occur.

Likewise, although all certain events have probability one, there are non-certain events that also have probability one (such as choosing a number that isn't $0.4858282$ in the previous set up). Such events are referred to as "almost never" or "almost always" or similar. So, a number selected uniformity at random from $(0,1)$ is almost never equal to $0.4858282$.

This question is another example of a possible, but probability zero event. The probability that you roll a $6$ at some point is $1$. However, it's certainly possible that you roll a $2$ every time and therefore never stop. In fact, you can build a bijection between the strings of die rolls and $[0,1)$ by writing down the string of rolls you get and interpreting $6$ as a $0$. Under this map, the set of possible outcomes where you don't terminate is the set of numbers that never contain a $0$. With some measure theory, this can be shown to be a set of measure $0$, and have probability $0$ of being chosen uniformity at random from $[0,1)$

Answer 2

You always have $5/6$ probability of not rolling a 6 in each roll. In $n$ rolls, the probability of not rolling a $6$ is $(\frac{5}{6})^n$ which is also non-zero for every $n$, but approaches zero as $n$ approaches infinity. Since realistically you cannot roll infinitely many times, just very often, it is possible to not roll a $6$ for any amount of rolls.