Expected average of number of tiles empty I have a grid of 3x4 and every tile on the grid has a corresponding chance of being empty. This chance is given. For example, consider the following grid:
$$ \frac{14}{15} \frac{1}{3} \frac{8}{13} \frac{1}{11} $$
$$ \frac{5}{6} \frac{7}{9} \frac{1}{10} \frac{4}{7} $$
$$ \frac{7}{11} \frac{4}{5} \frac{1}{12} \frac{1}{10} $$
I would like to know the expected number of empty tiles. I know that E[X] = P(X = 1) * 1 + P(X = 2) * 2 etc but how can I calculate this efficiently? For example, if this grid is 90x90, it would be a lot of work.
Thank you!

Just add up all the individual cell probabilities of being empty. Each cell's contribution to the expectation you are seeking is that cell's chance of being empty.
