Probability Outcomes and Models

A trial produces an outcome. The sample space is the set of all possible outcomes, and an event is a subset of that sample space. For equally likely outcomes, \(P(A)=\dfrac{\text{number of outcomes in}\hspace{0.5em}A}{\text{total number of outcomes}}\). Check equal likelihood before counting.

Probabilities satisfy \(0\le P(A)\le1\), \(P(S)=1\), and \(P(A^{\prime})=1-P(A)\). An impossible event has probability zero; a certain event has probability one.

Experimental relative frequency is \(\dfrac{\text{number of occurrences}}{\text{number of trials}}\). It estimates a probability and can vary between samples. For a fair die, the theoretical probability of an even score is \(3/6=1/2\); observing \(47\) even scores in \(100\) throws gives relative frequency \(0.47\).

A geometric area ratio represents probability only when the point is chosen uniformly over the region. For a point uniformly chosen in a square of area \(20\), an event region of area \(5\) has probability \(5/20=1/4\). A selection mechanism favouring one part of the square changes this calculation.