Discrete probability spaces
Probability on finite sets rests on counting. With equally likely outcomes, probability is favourable cases over total cases.
Sample spaces and events
A sample space \(S\) lists all outcomes; an event is a subset. With equally likely outcomes, \(P(E) = \frac{|E|}{|S|}\). Rolling two dice gives 36 outcomes; a sum of 7 occurs in 6 of them, so \(P = \frac{1}{6}\). Probabilities of all outcomes sum to 1.
P(E) = \frac{|E|}{|S|}Axioms of probability
Every probability satisfies three rules: \(P(E) \ge 0\); \(P(S) = 1\); and for disjoint events, \(P(E_1 \cup E_2) = P(E_1) + P(E_2)\). Everything else, such as the complement rule \(P(\overline{E}) = 1 - P(E)\), follows from these. Outcomes need not be equally likely: a loaded die simply assigns different probabilities.
Counting-based probability
The probability of a full house in poker is \(\frac{13\binom{4}{3} \cdot 12\binom{4}{2}}{\binom{52}{5}} = \frac{3744}{2{,}598{,}960} \approx 0.00144\). Counting the favourable outcomes carefully, by stages, is the whole challenge.
Conditional probability
\(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\): restrict the sample space to \(B\). If a card drawn is a face card, the probability it is a king is \(\frac{4}{12} = \frac{1}{3}\). Conditional probability is how information updates odds.
P(A \mid B) = \frac{P(A \cap B)}{P(B)}Independence
Events are independent when \(P(A \cap B) = P(A)P(B)\). Repeated coin flips are independent; draws without replacement are not. Independence greatly simplifies calculations, but must be justified, not assumed.
Bayes' theorem
\(P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}\) reverses conditioning. Spam filters compute the probability an email is spam given its words, using how often those words appear in known spam and non-spam emails. Computer scientists call these naive Bayes classifiers.
P(A \mid B) = \frac{P(B\mid A)P(A)}{P(B)}The Monty Hall problem
A prize is behind one of three doors. You pick door 1; the host, who knows where the prize is, opens another door showing no prize and offers a switch. Switching wins with probability \(\frac{2}{3}\): your first pick is right only \(\frac{1}{3}\) of the time, and the host's choice transfers the remaining \(\frac{2}{3}\) to the other closed door.
Common mistakes
Assuming outcomes are equally likely when they are not, such as sums of two dice. Confusing \(P(A \mid B)\) with \(P(B \mid A)\). Treating dependent events as independent. Trusting intuition in problems like Monty Hall without calculating.
Key takeaways
Probability follows three axioms. With equally likely outcomes, count favourable over total. Conditional probability restricts the sample space; independence means \(P(A \cap B) = P(A)P(B)\). Bayes' theorem reverses conditioning, and careful counting beats intuition.
Flashcard
All possible outcomes
Check your understanding
In the Monty Hall problem, what is the probability of winning if you switch?