So if there are two players playing a tile game where there are two sets of matching tiles $a_1, a_2, a_3$ and $b_1, b_2, b_3$, what would the optimal strategy be to maximize winning probability? Go first or second?
A turn consists of flipping over three of the cards to see if they match. If going first, then prob of winning is $1/6$? By going second, I can win $5/6$ of the time since I know that what he flipped over must be 2 from one set and 1 from another. Thus, flip any of the remaining three?
Player $1$ wins on the first turn with probability $\frac 1{10}$, needing to match the first tile, probability $\frac 25$, then again, probability $\frac 14$. Otherwise, player $2$ wins if his first tile matches the pair of the three that player $1$ flipped, probability $\frac 13$, or if he first flips a pair of player $1$'s singleton, probability $\frac 13$, for a total $\frac 23\cdot \frac 9{10}=\frac 3{5}$. The other $\frac 3{10}$ player $1$ wins on the second turn. So if you are player $1$, don't flip any tiles.