From an experiment to a sample space

A random experiment has an uncertain result, but its possible results can still be described precisely.

Write the sample space Ω before calculating. For one die, Ω = {1,2,3,4,5,6}.

An outcome is one member of Ω. An event is a set of outcomes, such as E = {2,4,6} for an even roll.

For equally likely outcomes, probability is favorable outcomes divided by all outcomes.

Start with the action, such as rolling a die, and separate it from the result. The action is the experiment; the single number that appears is one outcome.

A sample space must include every possible outcome exactly as finely as the question needs. For two coin tosses, {HH, HT, TH, TT} works, while merely writing {head, tail} loses the order and cannot answer questions about two tosses.

An event can contain one outcome, many outcomes, every outcome, or no outcomes. Thinking of an event as a set makes words such as 'or', 'and', and 'not' become union, intersection, and complement.

Fairness is what permits counting. If all eight spinner sectors have equal area, three winning sectors mean probability 3/8; if their areas differ, add the actual sector probabilities instead of counting labels.

Probability rules and complements

All probability mass is one whole pie. The part outside an event is whatever remains.

Every probability lies from 0 to 1, and P(Ω)=1.

The complement Aᶜ means A does not happen, so P(Aᶜ)=1−P(A).

For events that cannot happen together, add their probabilities. For overlapping events, subtract the overlap once.

Picture probability as one full litre of paint spread over the sample space. If an event uses 0.28 litre, its complement must use the remaining 0.72 litre because nothing can sit outside the whole space.

For 'A or B', first add both event probabilities. The shared region A∩B was then included twice, so subtraction removes exactly one duplicate and leaves every qualifying outcome counted once.

A quick reasonableness check catches many errors: a probability cannot be negative or exceed 1, a larger event cannot have smaller probability than an event inside it, and an event plus its complement must total 1.

This is used in monitoring systems by defining events such as 'server is slow' and 'server returns an error'. The union estimates how often users see either problem, while the intersection reveals requests suffering both, so engineers do not double-count incidents.

Practical deep dive for this lesson

Build and audit a real event model

A probability answer is trustworthy only when the experiment, outcomes, event, and weighting rule all describe the same real question.

Suppose a delivery app records each order as on time, late, or cancelled. Those three labels form a sample space only if every order receives exactly one label and no relevant outcome is missing.

To measure 'service problem', define the event {late, cancelled}; then estimate each outcome from comparable historical orders. If peak-hour orders behave differently, equal counting over labels is wrong and the sample should be split or weighted.

Finally check totals, time window, and denominator. Probability answers operational questions only when definitions are stable enough that yesterday's recorded frequencies represent the orders being planned for today.