PROBABILITY 01

Random Experiments

A random experiment has known possible outcomes, but the outcome of one trial is uncertain before it occurs.

The main idea

Probability begins with a repeatable situation in which uncertainty remains.

\[\boxed{\text{known possible outcomes}\; +\; \text{uncertain realised outcome}}\]

1. Deterministic versus random

If the same initial conditions always produce the same outcome, the model is deterministic. In a random experiment, repeated trials can produce different outcomes even when the procedure is unchanged.

2. Examples

Tossing a coin, rolling a die, counting infections during a time interval, or recording a measurement affected by noise can all be treated as random experiments.

3. Trial

One performance of the experiment is called a trial. Repeating the experiment produces a sequence of trials.

4. Sample space

The set of all possible outcomes is the sample space, usually denoted by \(\Omega\).

For a coin toss,

\[\Omega=\{H,T\}.\]

For a six-sided die,

\[\Omega=\{1,2,3,4,5,6\}.\]

5. Discrete and continuous outcome spaces

A finite or countable sample space is discrete. A measurement such as time, height or temperature may take values in a continuous range.

6. Probability comes after the experiment is defined

Before assigning probabilities, we must state clearly what the experiment is and what outcomes are possible.

\[\boxed{\text{experiment}\rightarrow\Omega\rightarrow\text{events}\rightarrow\text{probabilities}}\]

7. The central idea

A random experiment does not mean that everything is unknown. The possible outcomes and probability structure may be known even though the realised outcome is not.