Random Experiment, Sample Space and Algebra of Events

Every problem in this chapter begins with a random experiment — an experiment whose outcome cannot be predicted with certainty in advance, but whose full set of possible outcomes is known beforehand. Tossing a coin, throwing a die, or drawing a card from a well-shuffled pack are all random experiments: a single toss might land heads or tails, but which one is uncertain before the toss actually happens.

Basic terminology

• Trial: a single performance of a random experiment.
• Outcome: a possible result of one trial (e.g. 'Head' when a coin is tossed).
• Sample Space ($S$): the set of ALL possible outcomes of a random experiment. For one die, $S=\{1,2,3,4,5,6\}$, so $n(S)=6$; for two coins tossed together, $S=\{HH,HT,TH,TT\}$, so $n(S)=4$.
• Event: any subset of the sample space. E.g. 'getting an even number' on one die is the event $A=\{2,4,6\}$.
• Equally likely outcomes: outcomes with the same chance of occurring (an unbiased coin/die).
• Favourable outcomes: the outcomes of $S$ that belong to (favour) a given event.

Algebra of events — since every event is a set, ordinary set operations describe how events combine:

Two events are mutually exclusive if they share no common outcome (e.g. 'getting a 2' and 'getting a 5' on a single die throw); they are exhaustive if, taken together, they use up every outcome in $S$ (e.g. 'even number' and 'odd number' on a die are both mutually exclusive AND exhaustive at once). This distinction — mutually exclusive or not, exhaustive or not — is exactly what decides which version of the Addition Theorem (Section 3) applies to a given problem, so the Gujarat Std-12 Commerce Statistics syllabus expects a student to identify it correctly before reaching for any formula.