I would like to know if one can be absolutely certain, after a number of trials, to circle every integer from 1 to 100 using the following method:
Let's say you write down integers 0 to 100 from left to right. Then take one 6-sided die and roll it. The result gives the number of integers from 0 to move to the right along the number line you wrote down. Circle that number. Do it again, moving along the number line and circling numbers. So if on the first roll the die resulted in a 3, you move 3 spaces from zero to the number 3. If the second roll results in a 5, you move 5 spaces from the number 3, to the number 8. Then circle number 8. So now you have 3 and 8 circled. Do this all the way up through the numbers until the last die roll results in a number that would put you past 100. Call this set of results "layer 1".
Now repeat this again on the exact same set of numbers, with some already circled. This will be called "layer 2". Then do "layer 3" and so on until all numbers from 1 to 100 are circled (if possible).
What I am puzzled by is that common sense tells me that eventually all numbers from 1 to 100 will be circled. This appears to me to be absolutely certain. But the results depend on die rolls, which are based on probability, which says there's a small chance that 1 billion "layers" can still leave one number uncircled. I would guess that a resolution to this would be that there's a limit to how many layers of this there could be before all numbers are circled.
Or is there always going to be some probability that billions of layers of this can be done with still one or more numbers uncircled? I'm interested in seeing the mathematics behind this problem.
Let $k$ be an integer from $1$ to $100$. During the creation of any given layer $k$ has a chance of at least $\frac16$ of being circled (for most numbers probability is higher because there is more than one way to hit them). This means that the probability of $k$ being missed on a layer is at most $\frac56$.
So in $n$ layers, the probability of $k$ not being circled is at most $\left(\frac56\right)^n$. Since this expression goes to $0$ as $n\rightarrow\infty$, we would say that the probability of $k$ never being hit is $0$. This applies to every integer from $1$ to $100$.
So with probability $1$, all the numbers will eventually be circled.