Every Question About Chance Starts With a List of Outcomes

"What is the chance of an even number on a die?" can only be answered once you know every possible outcome and which of them count as even.

Probability uses the language of sets for this: all outcomes form one set, and every event is a subset of it.

This part covers sample spaces and events, the types of events, combining events, and mutually exclusive and exhaustive events.

**A random experiment has a set $S$ of all possible outcomes, called the sample space, and an event is any subset of $S$; the event occurs when the outcome belongs to that subset.

Worked example 1 — a die.** $S=\{1,2,3,4,5,6\}$. The event "an even number" is $E=\{2,4,6\}$. If a $4$ is rolled, $E$ has occurred.

Worked example 2 — two coins. $S=\{HH,HT,TH,TT\}$.

- "At least one head" $=\{HH,HT,TH\}$
- "Exactly one head" $=\{HT,TH\}$
- "No head" $=\{TT\}$

Worked example 3 — counting events. A sample space with $n$ outcomes has $2^n$ subsets, so it has $2^n$ possible events.

$$\text{one die: }{2}^6=64\text{ events}\text{two coins: }{2}^4=16\text{ events}$$

An everyday example. Rolling the die in a game of ludo: getting a $6$ to bring out a piece is the event $\{6\}$, a subset of $\{1,2,3,4,5,6\}$.

The substance. **$HT$ and $TH$ are different outcomes** when two distinct coins are tossed, so both must be listed.

**The empty set $\varnothing$ is the impossible event, the whole sample space $S$ is the sure event, a subset with exactly one outcome is a simple event, and a subset with more than one outcome is a compound event.

Worked examples with one die**, $S=\{1,2,3,4,5,6\}$:

- Impossible — "a number greater than $6$" $=\varnothing$
- Sure — "a number less than $7$" $=S$
- Simple — "a $5$" $=\{5\}$
- Compound — "a prime number" $=\{2,3,5\}$

With two coins. "Both heads" $=\{HH\}$ is simple; "at least one tail" $=\{HT,TH,TT\}$ is compound.

An everyday example. At the toss before a cricket match, "the coin shows heads" is a simple event, while "the coin lands heads or tails" is the sure event.

The substance. The type depends on how the sample space is written — with a die, "an even number" is compound because it contains three outcomes.

**Events combine exactly like sets: "not $A$" is the complement $A^{\prime }=S-A$, "$A$ or $B$" is the union $A\cup B$, "$A$ and $B$" is the intersection $A\cap B$, and "$A$ but not $B$" is $A-B=A\cap B^{\prime }$.

Worked example.** With one die, let $A$ = "an even number" $=\{2,4,6\}$ and $B$ = "a number greater than $3$" $=\{4,5,6\}$.

- **Not $A$**: $A^{\prime }=\{1,3,5\}$
- **$A$ or $B$**: $A\cup B=\{2,4,5,6\}$
- **$A$ and $B$**: $A\cap B=\{4,6\}$
- **$A$ but not $B$**: $A-B=\{2\}$
- **$B$ but not $A$**: $B-A=\{5\}$

With two coins. If $C$ = "at least one head" $=\{HH,HT,TH\}$, then $C^{\prime }=\{TT\}$, which is "no head".

An everyday example. A canteen offers a meal with rice or roti, and with dal or sabzi. "Rice and dal" is an intersection; "rice or dal" is a union that includes every plate with either.

The substance. "Or" in probability includes both — $A\cup B$ occurs when $A$ occurs, $B$ occurs, or both occur.

**Events are mutually exclusive when no two can happen together, so $A\cap B=\varnothing$; events are exhaustive when together they cover the sample space, so $E_1\cup E_2\cup \cdots \cup E_n=S$.

Worked examples with one die.**

- $\{2,4,6\}$ and $\{1,3,5\}$ — mutually exclusive and exhaustive
- $\{1,2\}$, $\{3,4\}$, $\{5,6\}$ — pairwise disjoint with union $S$, so mutually exclusive and exhaustive
- $\{1,2,3\}$ and $\{3,4,5,6\}$ — union $S$ but both contain $3$, so exhaustive, not mutually exclusive
- $\{1\}$ and $\{2\}$ — mutually exclusive, not exhaustive

A general fact. The simple events of any sample space are always mutually exclusive and exhaustive, and so are any event $A$ and its complement $A^{\prime }$.

An everyday example. A train arriving early, on time, or late — exactly one of these happens, so the three events are mutually exclusive and exhaustive.

The substance. For three or more events, every pair must be disjoint to call the whole collection mutually exclusive.