Probability — chapter notes

An experiment (tossing a coin, throwing a die, drawing a card) has a set of possible outcomes. An event is a collection of outcomes, e.g. "an even number" $= \{2, 4, 6\}$ for a die. An elementary event has exactly one outcome.

Outcomes are equally likely when each has the same chance: a fair coin or die, a well-shuffled deck, balls of the same size drawn without looking. The classical definition of probability needs equally likely outcomes.

For an experiment with equally likely outcomes,

• $0 \le P(E) \le 1$; an impossible event has $P = 0$, a sure event $P = 1$.
• The probabilities of all elementary events add up to $1$.

Example: one die: $P(\text{prime}) = \dfrac36 = \dfrac12$; $P(\text{greater than } 4) = \dfrac13$; $P(\text{multiple of } 7) = 0$.

"Not $E$" happens exactly when $E$ does not. So

Use it for "not", "at least one", "at most": e.g. at least one head with two coins $= 1 - P(\text{TT}) = \dfrac34$; at least one 6 with two dice $= 1 - \dfrac{25}{36} = \dfrac{11}{36}$.

• Coins: $n$ coins give $2^n$ outcomes: $2$, $4$, $8$. HT and TH are different. A tree diagram lists them.
• One die: $6$ outcomes.
• Two dice: $36$ ordered pairs $(a, b)$; use a $6 \times 6$ table. Sums are not equally likely: sum $7$ in $6$ ways, sums $2$ and $12$ in $1$ way each.

Examples: $P(\text{sum } 8) = \dfrac{5}{36}$; $P(\text{doublet}) = \dfrac16$; $P(\text{even product}) = \dfrac34$; three coins, $P(\text{exactly two heads}) = \dfrac38$.

A deck has $52$ cards in $4$ suits of $13$: spades ♠ and clubs ♣ (black), hearts ♥ and diamonds ♦ (red). Each suit has an ace, $2$–$10$, and the face cards jack, queen, king. So: $26$ red, $26$ black, $12$ face cards, $4$ of each rank.

• "$A$ or $B$": count each card once (spade or ace: $13 + 3 = 16$).
• Cards removed: recount the total first (no jacks: $48$ cards).

Examples: $P(\text{red king}) = \dfrac{1}{26}$; $P(\text{face card}) = \dfrac{3}{13}$; $P(\text{neither heart nor king}) = \dfrac{36}{52} = \dfrac{9}{13}$.

• Discs or tickets numbered $a$ to $b$: $b - a + 1$ outcomes. List multiples, squares or primes explicitly ($1$ is not prime). "Divisible by both $4$ and $6$" means divisible by $12$.
• Bags: each ball is an outcome (balls of the same size). $P(\text{colour}) = \dfrac{\text{balls of that colour}}{\text{total balls}}$. Adding or removing balls changes the total; unknown numbers lead to a simple equation.

Examples: discs $1$–$50$: $P(\text{square}) = \dfrac{7}{50}$; bag of $6$ red and $b$ blue with $P(\text{blue}) = \dfrac35$ gives $b = 9$.