Terminology

Definition 3.1.3 — Sample Space

The sample space of an experiment is the set of all possible outcomes. The uppercase letter $S$ denotes the sample space.

There are three standard ways to display a sample space: list the outcomes, draw a tree diagram, or draw a Venn diagram. If you flip one fair coin, $S = \{H, T\}$ — two outcomes, and the set of them is the sample space.

Definition 3.1.3: the sample space of two coin flips — every possible result appears in S exactly once.

Definition 3.1.4 — Event

An event is any combination of outcomes — that is, any subset of the sample space. Uppercase letters such as $A$ and $B$ represent events. The probability of an event $A$ is written $P(A)$.

The word combination is doing real work: an event does not have to be a single outcome. An event is a question you ask about the result — the answer is yes for some outcomes and no for the others.

Definition 3.1.4: an event is a subset of the sample space — the outcomes that answer yes are the event.

Definition 3.1.5 — Probability of an Outcome

The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive.

$P(A) = 0$ means $A$ can never happen; $P(A) = 1$ means $A$ always happens. Flip a coin 20 times, then 2,000, then 20,000, tracking the fraction of heads — it wanders early and settles onto 0.5.

Definition 3.1.5: every probability is a position on one line from 0 to 1.

Definition 3.1.6 — Equally Likely

Equally likely means that each outcome of an experiment occurs with equal probability.

When every outcome is equally likely, computing a probability becomes counting:

Toss a fair dime and nickel together: $S = \{HH, HT, TH, TT\}$. Let $A$ be exactly one head — $HT$ and $TH$ qualify, so $P(A) = \frac{2}{4} = 0.5$.

Definition 3.1.6: a fair die — six faces, six identical probabilities.

Definition 3.1.7 — Law of Large Numbers

As the number of repetitions of an experiment increases, the relative frequency obtained tends to become closer and closer to the theoretical probability.

The outcomes do not arrange themselves into a pattern to make this happen. Each trial is as unpredictable as the last — what changes is that the accumulated fraction has more trials underneath it, so any one strange run matters less.

Definition 3.1.7: the running fraction of heads settles onto 0.5 as the flips pile up.

Definition 3.1.8 — Biased

A coin, die, or other chance device is biased when its outcomes are not equally likely — that is, when at least one outcome occurs with a different probability than another.

Two math professors had their students test the Belgian one-euro coin. In 250 trials, heads came up 56% of the time and tails 44% — though 250 trials is not many, and more repetitions would be needed to draw a firm conclusion.

Definition 3.1.8: the Belgian euro coin came out 56/44 over 250 trials, not the 50/50 a fair coin predicts.

Definition 3.1.9 — “OR” Event

An outcome is in $A$ OR $B$ if it is in $A$, in $B$, or in both. Also written $A \cup B$ (union). Unlike everyday English, mathematics includes the overlap.

Let $A = \{1,2,3,4,5\}$, $B = \{4,5,6,7,8\}$. Then $A \cup B = \{1,2,3,4,5,6,7,8\}$ — 4 and 5 are not listed twice; a set records what is in it, not how many ways an element got there.

Definition 3.1.9: A OR B covers both circles, and the overlap is shaded once, not twice.

Definition 3.1.10 — “AND” Event

An outcome is in $A$ AND $B$ if it is in both $A$ and $B$ at the same time. Also written $A \cap B$ (intersection).

Using the same two sets from Definition 3.1.9, $A \cap B = \{4, 5\}$ — only the elements that appear in both lists.

Definition 3.1.10: A AND B is only the lens — the outcomes in both sets at the same time.

Definition 3.1.11 — Complement

The complement of event $A$, denoted $A'$, consists of all outcomes NOT in $A$. Because every outcome is in $A$ or $A'$ and never both,

Let $S = \{1,\ldots,6\}$, $A = \{1,2,3,4\}$. Then $A' = \{5,6\}$, and $\frac{4}{6} + \frac{2}{6} = 1$.

Definition 3.1.11: A and its complement tile the whole sample space, so their probabilities add to 1.

Definition 3.1.12 — Conditional Probability

$P(A \mid B)$ is the probability $A$ occurs given that $B$ already has. Conditioning reduces the sample space to $B$:

Roll a die: let $A$ be 2-or-3, $B$ be even, so $B = \{2,4,6\}$. Inside $B$, only 2 qualifies, so $P(A \mid B) = \frac{1}{3}$ — by the formula, $\frac{1/6}{3/6} = \frac{1}{3}$ too.

Definition 3.1.12: conditioning on B discards the outcomes outside B, so the denominator drops from 6 to 3.

Find $P(M)$, $P(W)$, $P(T)$, $P(C)$, $P(M \text{ AND } T)$, $P(W \text{ AND } C)$, $P(M \text{ OR } W)$, $P(M')$, and the conditionals.

Margins: 0.48, 0.52, 0.38, 0.62. Cells: $P(M \text{ AND } T)=0.22$, $P(W \text{ AND } C)=0.36$. $P(M \text{ OR } W)=1$, $P(M')=0.52$. $P(T\mid M)=\frac{22}{48}\approx 0.4583$, $P(C\mid W)=\frac{36}{52}\approx 0.6923$.

$M$=male, $F$=female, $R$=right-handed, $L$=left-handed. Find the margins, the AND cells, the OR unions, $P(M')$, and the conditionals.

$P(M)=0.52$, $P(F)=0.48$, $P(R)=0.87$, $P(L)=0.13$. $P(M \text{ AND } R)=0.43$. $P(R\mid M)=\frac{43}{52}\approx0.8269$, $P(L\mid F)=\frac{4}{48}\approx0.0833$ — the same cell, 4, divided by two different reduced sample spaces.