Perhaps a better analogy is this: Imagine six six-sided dice, and six boxes labeled 1 to 6 on the table in front of you*; i.e. one die per box into which it will be rolled. Assuming the die rolls are independent of one another, the probability of each die showing the label of its box is 1/6 - no matter what that label is! The probability of one die showing its box's label and also a second die independently showing its box's label is (1/6)*(1/6); N dice, (1/6)^N.
However, this all changes if the die rolls are not truly random or not truly independent of one another!
* That's just for simplicity; it's too easy to get hung up on those numbers - the number of dice, boxes, and sides, or even whether they are labeled with numbers or hieroglyphs does not matter!