Random Experiments and Sample Space

Probability is the branch of mathematics that measures how likely an uncertain event is to happen. Before we can measure that likelihood we must describe the situation precisely, and that begins with the idea of a random experiment.

A random experiment is a process that can be repeated any number of times under the same conditions, whose every possible outcome is known in advance, but whose actual outcome on any single trial cannot be predicted with certainty. Tossing a coin, rolling a die, drawing a card from a well-shuffled pack, or picking a ticket from a box are all random experiments — we know the list of possible results, yet not which one will occur.

Each individual result of a random experiment is called an outcome (or a sample point). The set of all possible outcomes is the sample space, denoted $S$ (some books write $U$ or $\Omega$). Every random-experiment analysis starts by writing down $S$ correctly and counting the outcomes in it, written $n(S)$.

When a coin is tossed $n$ times (or $n$ coins once), $n(S) = 2^{n}$; when a die is rolled $n$ times, $n(S) = 6^{n}$. Counting techniques from the Permutations and Combinations chapter are used constantly here, because a probability can only be as reliable as the count of outcomes behind it. This treatment follows the same probability principles set out in the standard national mathematics curriculum, on which the Maharashtra Std XI Mathematics and Statistics syllabus is based.