Expected average of number of tiles empty

Question

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!

Answer 1

Define the variable $X_i$ taking value $1 $ if cell $i$ is empty, $0$ otherwise. Then $E(X_i)=p_i$ with $p_i=$ probability of cell being empty. Then $Z=\sum X_i$ is the number of empty cells, and $E(Z)=\sum E(X_i) = \sum p_i$. I.e., we just sum up the numbers of the grid. Notice that, because linearity of expectation, we don't need to know if $X_i$ are independent.

Answer 2

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.