Of the 36 possible theoretical rolls of two dice: 25 rolls show no 2 on each: 10 rolls show only one 2 on each; and 1 roll shows two 2's.
When at least one die shows 2, for every 10 rolls of only one 2 each there will be 1 roll of both dice showing a 2. That means one in eleven times overall there will be a double 2. (Note that the required single 2 may be on either die. If the 2 is on the die on the left, then there are five non-2 possibilities for the die on the right; and if the 2 is on the die on the right, then there are five non-2 possibilities for the die on the left.)
However, we don't have to think like mathematicians. I maintain that a better answer is one in six times. The condition that "at least one die is a 2" is the same as "one or more of the dice is a 2". Hence, the only way to properly and fully meet that condition in terms of two dice by consecutive rolls would be to look at only one die first. Were it a 2, then would have the "one a 2", and possibly the "or more a 2" by the other die. But, were it not a 2, then not possibly the "or more a 2" by the two dice in the roll. Two dice must be considered separately, ie, not as one possible outcome together of a (simultaneous) set of such possible outcomes.
Note that in the sentence, "Of the 36 possible theoretical rolls of two dice: 25 rolls show no 2 on each: 10 rolls show exactly one 2 on each; and 1 roll has both dice showing a 2 at the same time," the operative word is "and". Writing "or" instead of "and" would be the mathematical approach to the condition "at least one die is a 2"; but not the physical, real-life approach to this exercise in general.
