Probability - Random experiments and sample space

Random Experiment: A process or operation where the result cannot be predicted with certainty even though all possible outcomes are known in advance. For example, tossing a fair coin is a random experiment because you know it will land on heads or tails, but you don't know which one will occur in a specific trial.

Sample Space ($S$): The set of all possible outcomes of a random experiment. Visually, imagine a universal set containing distinct points, where each point represents one possible result. For a single die roll, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$, representing the six faces of the cube.

Event ($E$): Any subset of the sample space. An 'Elementary Event' is an outcome that cannot be further decomposed (like rolling a $4$), while a 'Compound Event' consists of more than one outcome (like rolling an even number: $\{2, 4, 6\}$).

Equally Likely Outcomes: Outcomes are considered equally likely if no particular outcome is more expected to occur than any other. In a fair experiment, such as spinning a wheel divided into equal colored sectors, each sector has the same geometric area and thus the same chance of the pointer landing on it.

Probability of an Event: A numerical measure of the likelihood that an event will occur, expressed as the ratio of favorable outcomes to the total number of outcomes. Visually, this can be seen as the 'size' of the event subset relative to the 'size' of the entire sample space box.

Impossible and Sure Events: An event that can never happen has a probability of $0$ and is called an impossible event (e.g., drawing a blue ball from a bag containing only red balls). An event that is certain to happen has a probability of $1$ and is called a sure or certain event.

Complementary Events: For any event $A$, the event 'not $A$' (denoted as $\bar{A}$ or $A'$) consists of all outcomes in the sample space that are not in $A$. Visually, if $A$ is a circle inside the sample space rectangle, $\bar{A}$ is the entire shaded area outside that circle but within the rectangle.

Multiple Trials and Sample Space Size: When multiple independent trials occur, the sample space grows exponentially. For $n$ coins, there are $2^n$ outcomes; for $n$ dice, there are $6^n$ outcomes. A tree diagram is often used to visualize these outcomes, where each branch represents a possible choice at each step.