Probability Language and Basic Rules

The sample space \(S\) is the set of all possible outcomes of a random experiment. An individual outcome is a sample point; an event \(A\) is a subset of \(S\).

The whole space \(S\) is the certain event, and \(\varnothing\) is the impossible event. The complement \(A^{\prime}\) contains the outcomes for which \(A\) does not occur.

\(0\le P(A)\le1,\qquad P(\varnothing)=0,\qquad P(S)=1\).

\(P(A^{\prime})=1-P(A),\qquad\varnothing^{\prime}=S,\qquad S^{\prime}=\varnothing\).

For equally likely outcomes in a finite sample space,

\(P(A)=\dfrac{n(A)}{n(S)}\), where \(n(A)\) counts the favourable outcomes and \(n(S)\) counts all outcomes.

From repeated trials under the same conditions, estimate \(P(A)\) by the relative frequency \(\dfrac{\text{number of occurrences of}\hspace{0.5em}A}{\text{number of trials}}\). The estimate can fluctuate from one set of trials to another.

When a point is chosen uniformly from a region of finite, nonzero length, area or volume,

\(P(A)=\dfrac{\text{measure of the favourable region}}{\text{measure of the whole region}}\). Use the same measure and units in numerator and denominator. This ratio does not apply to non-uniform selection.