# Probability of Rolling a number that changes over time on One Die

I've seen the following question asked and answered many times on various forums:

Knowing that the odds of rolling a six with one die are one in six, what are the odds of rolling a six in two, three, four (etc.) consecutive rolls? 1/6 + 1/6 + ... wouldn't make sense as you'd have a 100% chance of rolling a six in six rolls.

The answer I've seen is that the odds of rolling a six after n rolls are 1 - (5/6)^n.

Now my question is this - lets say that instead of rolling for a six, we pick a number out of a fixed sequence of six unique numbers after each roll. So for example, lets say our sequence is 1, 2, 3, 4, 5, 6. At the first roll we're looking to roll a one, the second roll we're looking for a two, then a three etc. until finally at the seventh roll we go back to one. What are the chances of matching any of the numbers in the sequence in this way after n rolls?

Yes, if I understand your question correctly. In each roll there is one out of six numbers you consider a match, whether it is a six or any other number. Thus, $$P\{\text{At least one match in n rolls}\}=1-P\{\text{No match in n rolls}\}=1-P\{\text{No match in roll 1, no match in roll 2...}\}=1-(5/6)^n,$$ where the last step follows from independence of the rolls.