Chapter 14 — Probability
Chance has rules. Describe every possible result of an experiment as a set, build events with “or”, “and” and “not”, and measure them with a probability that obeys three simple axioms — including $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
In seventeenth-century Paris a keen gambler, the Chevalier de Méré, noticed something odd. He had made money for years betting that he would get at least one six in four throws of a die. So he tried what seemed the same bet scaled up: at least one double six in twenty-four throws of a pair of dice. His reasoning was simple — a double six is six times rarer than a six, so six times as many throws ($4\times6=24$) should give the same chance. Yet he slowly lost money. He took the puzzle to Blaise Pascal, and Pascal's letters with Pierre de Fermat about it (and about how to split the stakes of an unfinished game fairly) are usually taken as the birth of the mathematics of probability. The answer, which you will be able to work out yourself by the end of this chapter, is that the first bet wins with probability $1-\left(\tfrac56\right)^4\approx0.518$ and the second only with probability $1-\left(\tfrac{35}{36}\right)^{24}\approx0.491$. The difference is tiny, but over hundreds of games it decides who goes home richer.
Today the same mathematics sets insurance premiums, judges whether a new vaccine works better than chance, predicts the chance of rain, designs fair lotteries and tells a bank how hard a six-digit one-time password is to guess. In Class 10 you met probability as "favourable outcomes divided by total outcomes". In this chapter you will make that idea precise. First you describe every possible result of an experiment as a set, the sample space, so that every event becomes a subset and the set operations of Chapter 1 become the words "or", "and" and "not". Then you will meet the axiomatic approach: three short rules that every probability must obey. From them follow all the familiar facts — that a probability lies between $0$ and $1$, that $P(\text{not }A)=1-P(A)$, the formula $\tfrac{n(E)}{n(S)}$ for equally likely outcomes, and the addition rule for "$A$ or $B$" — together with the counting tools of Chapter 6 for experiments too large to list.
- write the sample space of an experiment (coins, dice, cards, selections), and count its points with the multiplication principle, ${}^nP_r$ or ${}^nC_r$ when it is too large to list;
- describe an event in words and as a subset of the sample space, and recognise simple, compound, sure and impossible events;
- form the complementary event $A'$ and the events "$A$ or $B$", "$A$ and $B$" and "$A$ but not $B$", and decide whether events are mutually exclusive, exhaustive, or both;
- state the axioms of probability, check whether an assignment of probabilities to the sample points is valid, and deduce $P(\phi)=0$, $0\le P(E)\le1$ and $P(A')=1-P(A)$;
- find probabilities of events with equally likely outcomes, including counting-based problems on committees, cards and arrangements;
- use the addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ and its relatives to find the probabilities of "or", "neither", "exactly one" and "$A$ but not $B$".
Sections in this chapter
Events and Their Algebra
The sample space, events as subsets, simple, compound, sure and impossible events, the complementary event, "or", "and", "but not", and mutually exclusive and exhaustive events.
The Axiomatic Approach to Probability
Three axioms, probabilities of sample points, the complement rule, equally likely outcomes, counting with ${}^nC_r$, and the addition rule for "or".