Random Experiments and Outcomes

Random Experiments and Their Outcomes

In everyday language, an experiment often means a scientific test carried out under controlled conditions. In probability theory the word is used more broadly: an experiment is any action or process that produces some observable result. Tossing a coin, rolling a die, drawing a card from a shuffled deck, or measuring the lifetime of a bulb are all experiments in this sense.

Deterministic vs. Random Experiments

An experiment is called deterministic if, whenever it is repeated under identical conditions, it always produces the same result. For example, if water is heated to $100^\circ\text{C}$ at sea-level atmospheric pressure, it always boils — there is no uncertainty about the outcome.

An experiment is called a random experiment if:

1. it has more than one possible outcome, and
2. it is not possible to predict which particular outcome will occur, even though every possible outcome is known beforehand.

Tossing a fair coin is the standard example: we know in advance that the result will be either Head (H) or Tail (T), but we cannot say which one will occur on any particular toss. Rolling a die, drawing a card, and picking a ball from a bag of mixed-colour balls are all random experiments in exactly this sense.

Outcomes and Trials

Each time a random experiment is actually performed, it is called a trial, and the result obtained is called an outcome. When a coin is tossed once (one trial), the outcome is either H or T. When a die is thrown once, the outcome is one of $1, 2, 3, 4, 5, 6$.

The whole of probability theory rests on being able to list every possible outcome of a random experiment exhaustively and without repetition. This complete listing is called the sample space, and it is developed formally in the next section.