I was trying to come up with a dice game which makes the players' intuition fail.
The game has two dice. Player $A$ and player $B$ have different victory conditions. Player $A$ and $B$ alternate tossing the dice. Player $A$ needs to throw two consecutive $6$ to win. Player $B$ throws both dice together and needs to roll double sixes.
The expected number of rolls for player $B$ before winning is $36$. For player $A$, it is $42$. Should this be enough to convince someone to play as player $B$?