Summary

This chapter developed probability theory involving dependence between events and its application to random variables. Conditional probability, $P(A\mid B)=P(A\cap B)/P(B)$, measures the probability of $A$ once $B$ is known to have occurred, and rearranging its definition gives the multiplication theorem, $P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$ -- the natural tool for "without replacement" problems. Two events are independent when $P(A\cap B)=P(A)P(B)$, equivalently $P(A\mid B)=P(A)$; independence and mutual exclusivity (for events of positive probability) can never hold simultaneously. When an event $A$ can occur through several mutually exclusive and exhaustive routes $E_1,\dots,E_n$ (a partition of the sample space), the total probability theorem gives $P(A)=\sum_i P(E_i)P(A\mid E_i)$, and Bayes' theorem reverses this to find the posteriori probability of a particular cause given that $A$ was observed: $P(E_i\mid A)=\dfrac{P(E_i)P(A\mid E_i)}{\sum_j P(E_j)P(A\mid E_j)}$. A random variable is a real-valued function on the sample space, and its probability distribution lists every value against its probability, subject to $p_i\ge0$ and $\sum p_i=1$. Finally, the mean $E(X)=\sum x_ip_i$ gives …