Expected Value: Checkers The first checker is placed randomly in one of the eight positions on the diagonal shown here: ![Checker Diagram]( Checkers are then placed in the squares that lie below and to the right of the first checker, including those squares that are directly below or directly to the right of the first checker. What, then, is the expected number of checkers on the board? * * * Listing it all out won't be efficient, maybe cases? But there's too many cases. Thanks in advanced for answering!

As was stated in the comments you just need $\frac{1}{8}\sum_{i=1}\limits^8i^2$. (do you see why?)

If you don't want to add $8$ numbers you can use the formula $\sum\limits_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}$.

When $n=8$ you get $\frac{1}{8}\frac{8\times9\times 17}{6}=\frac{3\times 17}{2}=25.5$