Q.Two coins are tossed simultaneously. Write the sample space $S$ and state $n(S)$. Also write the event $A$: "at least one head appears" and find $n(A)$.

Experiment. An experiment is any process whose result is well defined. It is deterministic when the outcome can be predicted with certainty under ideal conditions (e.g. genetic determination, hitting a fixed target under fixed conditions). It is a random (non-deterministic) experiment when (i) every possible outcome is known in advance, (ii) no individual outcome can be predicted in advance, and (iii) it can be repeated under identical conditions -- rolling a die and tossing a coin are the standing examples.

Sample space and elementary events. The sample space $S$ of a random experiment is the set of ALL its possible outcomes; each point of $S$ is called a simple event, elementary event, or sample point, and cannot be broken down any further. $S$ can be:

• finite -- $S=\{1,2,3,4,5,6\}$ for a die, $S=\{H,T\}$ for a coin;
• countably infinite -- tossing a coin until the first head gives $S=\{H,TH,TTH,TTTH,\ldots\}$, which can be listed and put in one-to-one correspondence with $\mathbb N$;
• uncountably infinite -- choosing a real number $x$ with $0<x<1$ gives $S=\{x:0<x<1\}$, which cannot be listed at all.

This chapter, from here on, restricts to a finite sample space.

Events. When $S$ is finite, ANY subset of $S$ is an event -- formally, every element of the power set $\mathcal P(S)$ is an event. Named types:

• Sure (certain) event -- $S$ itself (it always happens).
• Impossible event -- the empty set $\varnothing$ (it never happens).
• Complementary event $\bar A$ (also written $A^c$ or $A'$) -- 'not $A$', the event that $A$ does not occur.
• Mutually exclusive events -- $A_1,\ldots,A_k$ with $A_i\cap A_j=\varnothing$ for every $i\ne j$: they cannot occur simultaneously.
• Mutually inclusive events -- $A_i\cap A_j\ne\varnothing$ for some $i\ne j$: they CAN occur simultaneously.
• Exhaustive events -- $A_1\cup A_2\cup\cdots\cup A_k=S$: between them they cover every outcome.
• Mutually exclusive AND exhaustive -- both conditions together; this is exactly a partition of $S$, the structure Total Probability and Bayes' Theorem are built on.
• Equally likely events -- events with the same chance of occurring (rolling a fair die gives six equally likely faces; a loaded/coloured die need not).

Notation. For events $A,B$: $A\cup B$ means '$A$ or $B$ or both'; $A\cap B$ (also written $AB$) means '$A$ and $B$ simultaneously'; $\bar A$ (or $A'$, $A^c$) means '$A$ does not occur'; and $A\cap\bar B$ means 'only $A$ occurs (not $B$)'.