Choosing which consecutive die pair to wait for

(Answer by intuition, without writing anything down.) A fair die is rolled repeatedly until a chosen ordered pair of values appears on two consecutive rolls, and you are paid one unit per roll made. Which target should you prefer: (1) a $4$ followed by a $5$, (2) a $4$ followed by another $4$, or (3) does it make no difference? Give the case number.

Case 2. Both targets need a 4 first. In the 4-then-5 game, a failed attempt that rolls another 4 still leaves you one roll from finishing, whereas in the 4-then-4 game every failure after a 4 sends you back to the start. So 4-4 takes longer on average (42 rolls versus 36), which is what you want when paid per roll.