Sample Spaces, Events and the Axioms

First, list what can hap­pen#

Before a cricket match the cap­tains toss a coin, and every­one agrees it is a fair way to decide. Prob­a­bil­ity starts to become pre­cise the moment you write down every out­come that could pos­si­bly hap­pen. So we begin with ran­dom exper­i­ments, sam­ple spaces and events. Then we learn to com­bine events with "and", "or", "not" and "but not", see what mutu­ally exclu­sive and exhaus­tive events are, and fin­ish with the few axioms every prob­a­bil­ity must obey.

Ran­dom exper­i­ments and sam­ple spaces#

A ran­dom exper­i­ment is one that has more than one pos­si­ble out­come, so you can­not pre­dict exactly what will hap­pen. Its sam­ple space $S$ is the set of all pos­si­ble out­comes, and each sin­gle out­come is called a sam­ple point.

• Toss two coins: $S = \{HH, HT, TH, TT\}$.
• Roll a die and toss a coin: $S = \{1H, 1T, 2H, \ldots, 6T\}$, $12$ points.
• Toss a coin until a head appears: $S = \{H, TH, TTH, \ldots\}$, infi­nite.
• Draw two balls from a bag of $2$ red and $3$ blue. You can describe $S$ by colours, $\{RR, RB, BR, BB\}$, or by the indi­vid­ual balls, $^5C_2 = 10$ points.

Events#

An event is a sub­set of $S$. We say it occurs when the out­come we actu­ally get belongs to it. $\varnothing$ is the impos­si­ble event, $S$ is the sure event, and an event with just one point is called sim­ple.

Com­bin­ing events. $A \cup B$ means $A$ or $B$ (or both). $A \cap B$ means both hap­pen. $A'$ (not $A$) $= S - A$. And $A - B = A \cap B'$ means $A$ but not $B$.

Mutu­ally exclu­sive: $A \cap B = \varnothing$, so the two can never hap­pen together. Exhaus­tive: $A_1 \cup \cdots \cup A_n = S$, so at least one of them is bound to hap­pen.

Axioms of prob­a­bil­ity#

A prob­a­bil­ity $P$ on a finite sam­ple space gives each event a num­ber, fol­low­ing three sim­ple rules:

1. $P(E) \ge 0$ for every event $E$;
2. $P(S) = 1$;
3. if $E$ and $F$ are mutu­ally exclu­sive, $P(E \cup F) = P(E) + P(F)$.

Every­thing else fol­lows from these. For instance $P(\varnothing) = 0$, $P(E) \le 1$, and $P(E)$ is the sum of the prob­a­bil­i­ties of its sam­ple points. When all $n$ out­comes are equally likely, $\displaystyle P(E) = \tfrac{n(E)}{n(S)}$.

Try these your­self#

1. Write down the sam­ple space for each: (a) a coin tossed three times; (b) a die rolled twice (how many points?); (c) a fam­i­ly's two chil­dren by sex, older first.
2. A coin is tossed and, if it shows a head, a die is rolled. Describe $S$.
3. Two dice are rolled. List: $A$ = sum is $8$; $B$ = both show the same num­ber; $C$ = first shows $5$. Find $A \cap B$, $A \cap C$, $B \cup C$ in terms of counts.
4. Which pairs from ques­tion 3 are mutu­ally exclu­sive?
5. A card is drawn from a pack. $A$ = heart, $B$ = king, $C$ = red. Describe $A \cap B$, $A \cup C$, $C - A$.
6. Check whether $\displaystyle P(a) = \tfrac{1}{3}$, $\displaystyle P(b) = \tfrac{1}{2}$, $\displaystyle P(c) = \tfrac{1}{4}$ is a valid assign­ment on $\{a, b, c\}$.
7. A coin is biased so that heads is three times as likely as tails. Find $P(H)$ and $P(T)$.
8. In the loaded die of Exam­ple 3, find $P(\text{prime})$.
9. Give an exam­ple of three events that are mutu­ally exclu­sive but not exhaus­tive.
10. Show that $A$ and $A'$ are always mutu­ally exclu­sive and exhaus­tive.

Answers to check against#