I have 3 regular dice with six sides. I am playing a game where I win when I match all three dice to the same side. My expected shannon bits of entropy H(X) evaluates to 7.754887502 bits. However, my dice aren't very good as a random number generator and the entropy is actually measured at 6.965784285. How do I recalculate my new theoretical win rate?
$$ P_{expected} = (\frac 1 6) ^ 2 $$
First, I believe $P_{expected}=(\frac{1}{6})^2$ because it doesn't matter what the first die is, just that the other 2 match. The value you give for $P_{expected}$ is the value for a specific value, say, 6. There are six such values, so $(\frac{1}{6})^3•6=(\frac{1}{6})^2$.
Second, there is a difference between theoretical probability and experimental probability. I wouldn't recalculate my theoretical probability; I would re-run my experiment again (and again and again), updating the experimental probability by averaging the results. If your experimental probability continues to differ significantly from your theoretical probability, I wouldn't change the theoretical model; I would get better dice. :)