Q.Find the total number of outcomes when 5 coins are tossed once.

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. …