PROBABILITY 03
Probability
Probability assigns a number between zero and one to an event and measures how likely that event is under a specified model.
The probability scale
A probability of \(0\) represents impossibility in the model; probability \(1\) represents certainty.
1. Equally likely finite outcomes
If \(n\) outcomes are equally likely and \(m\) belong to event \(A\), then
\[P(A)=\frac{m}{n}.\]2. Example
For a fair die, the event \(A=\{2,4,6\}\) has
\[P(A)=\frac36=\frac12.\]3. Addition rule
For any two events,
If the events are disjoint, the intersection term is zero.
4. Complement rule
\[\boxed{P(A^c)=1-P(A)}.\]5. Relative frequency
When an experiment is repeated many times, the observed relative frequency of an event often approaches its model probability:
\[\frac{\text{number of occurrences of }A}{\text{number of trials}}.\]Probability is the theoretical quantity; relative frequency is an empirical estimate.
6. Probability depends on a model
A probability statement is meaningful only after the experiment, event and probability assumptions are specified.
7. Probability over a small time interval
If an event occurs continuously in time at rate \(\lambda\), then for sufficiently small \(\Delta t\),
\[P(\text{event in }\Delta t)\approx\lambda\Delta t.\]The rate and probability are different quantities: the probability is approximately rate multiplied by interval length.