Laws of Probability

Total Probability

The sum of probabilities is one.

Additivity

You can add probabilities.

Large Numbers

As you repeat something several times, the average will converge to the expected value.

Independence

If $x$ and $y$ are independent variables, $f(x)$ and $g(y)$ are independent.

Linearity of Expectations

The expected value of a sum of random variables is the same as the sum of the individual expected values, regardless of whether the variables are independent or dependent.

\[E[ax+b] = aE[x] + b\]