Business Mathematics and Basic Statistics · Ch 11 — Introduction to Probability

Random Experiment and Sample Space

In everyday life we constantly meet processes whose exact result cannot be predicted in advance, even though every result that COULD occur is already known beforehand — tossing a coin, rolling a die, or drawing a card from a well-shuffled deck are all examples. A process of this kind is called a random experiment: a trial that can, in principle, be repeated under identical conditions, whose individual outcome is uncertain, but whose complete list of possible outcomes is known in advance.

The set of ALL possible outcomes of a random experiment is called its sample space, denoted $S$ . Each individual outcome inside $S$ is called a sample point, and the number of sample points is written $n(S)$ .

Some standard sample spaces used throughout this chapter:

Tossing a single fair coin once: $S = \{H, T\}$ , so $n(S) = 2$ .
Rolling a single fair die once: $S = \{1, 2, 3, 4, 5, 6\}$ , so $n(S) = 6$ .
Drawing one card from a well-shuffled deck of 52 playing cards: $n(S) = 52$ (13 cards — Ace to King — in each of 4 suits: hearts, diamonds, clubs, spades).

When a random experiment is really TWO (or more) simpler experiments performed together — for instance, tossing a coin AND rolling a die at the same time — the combined sample space is built by pairing every outcome of the first experiment with every outcome of the second, and the total count of sample points multiplies:

n(S) = n(S_1) \times n(S_2)

for a first stage with $n(S_1)$ outcomes and a second, independent stage with $n(S_2)$ outcomes. This is called the multiplication principle of counting, and it is the standard tool for writing down a sample space too large to be obviously listed by hand.

Note

A Random Experiment vs. a Predictable Process

Not every repeatable process is "random" in this technical sense. If a process always produces the SAME result under the same conditions (e.g. adding 2 and 3 always gives 5), it has no sample space of multiple outcomes to speak of — the defining feature of a random experiment is genuine uncertainty in the individual outcome, combined with certainty about the full list of what could happen.

This WBCHSE Class 11 Business Mathematics and Basic Statistics (BMBS) chapter builds the vocabulary and the classical probability formula from exactly these everyday random experiments — tossing coins, rolling dice, and drawing playing cards — because their sample spaces are small, exact, and easy to list completely, which makes them the clearest setting to learn the underlying ideas correctly.

Definition 1Random experiment

A trial or process that can be repeated under identical conditions, whose individual outcome cannot be predicted with certainty in advance, but whose set of all possible outcomes IS known beforehand (e.g. tossing a coin, rolling a die, drawing a card).

Definition 2Sample space ($S$) and sample point

The sample space $S$ is the set of ALL possible outcomes of a random experiment; each individual outcome inside $S$ is a sample point. $n(S)$ denotes the number of sample points (e.g. $n(S)=6$ for a single die roll).

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