A battleship game is played on a $4 \times 4$ matrix, player one can place their domino(that takes up $2$ adjacent spaces) in one of $24$ places(There are $3$ places in each row and each column it can be placed so $3\cdot 4+3 \cdot 4=24$). Player $1$ secret places the domino and player $2$ secret guesses, which one of the $16$ indexes it's in. If player $2$ guess right they win $1$ dollar and player $1$ loses $1$ dollar. If player $2$ guesses wrong then player $1$ wins $1$ dollar and player $2$ loses $1$ dollar. The expected payoff for player $1$ is $\frac{3}{4}$. Prove this is true.
Reducing this to a 3x4 payoff matrix
$\begin{bmatrix}2&2&2\\2&2&2\\2&2&2\\2&2&-2\end{bmatrix}$
The above is an example of if player 1 being the row picked a corner then p2 picked the same corner(not entirely sure this is done correctly)
No need to count the number of arrangements. Two out of $16$ squares are covered, so the probability to hit a covered square is $\frac2{16}=\frac18$. The expected payoff follows directly.