Probability - Classical Definition of Probability

Random Experiment and Outcomes: A random experiment is a process where the result cannot be predicted with absolute certainty. The possible results of such an experiment are called outcomes. Visually, you can imagine a tree diagram where the starting point is the action (like flipping a coin) and the branches represent the possible outcomes (Heads or Tails).

Sample Space: The set of all possible outcomes of a random experiment is called the sample space, denoted by $S$. Imagine a rectangular 'Venn Diagram' frame containing several distinct points; each point represents a unique outcome, and the entire area inside the frame is the Sample Space.

Equally Likely Outcomes: Outcomes are said to be equally likely if none of them is expected to occur in preference to the others. For example, in a fair six-sided die, each face from $1$ to $6$ has the exact same physical area and weight, ensuring each number has a $\frac{1}{6}$ chance of appearing.

Theoretical Probability: Also known as Classical Probability, it is calculated based on the assumption that all outcomes are equally likely. It is the ratio of the number of favorable outcomes to the total number of outcomes. Visually, if the sample space is a pie chart, the probability of an event is the 'slice' of the pie that represents the favorable outcomes.

Range of Probability: The probability of an event $E$ is always a value such that $0 \le P(E) \le 1$. On a probability number line, $0$ indicates an 'Impossible Event' (like rolling a $7$ on a standard die) and $1$ indicates a 'Sure Event' (like rolling a number less than $7$ on a standard die).

Complementary Events: For every event $E$, there exists an event 'not $E$' (denoted as $\bar{E}$ or $E'$) which represents $E$ not occurring. If you visualize a circle representing the Sample Space, and a smaller circle inside it is $E$, then everything outside the small circle but inside the large circle is $\bar{E}$.

Elementary Events: An event that consists of only one outcome of the experiment is called an elementary event. A key property is that the sum of the probabilities of all the elementary events of an experiment is exactly $1$.

Standard Deck of Cards: A standard deck contains $52$ cards, split into $4$ suits: Spades (♠), Hearts (♥), Diamonds (♦), and Clubs (♣). There are $26$ red cards (Hearts and Diamonds) and $26$ black cards (Spades and Clubs). Each suit has $13$ cards: Ace, $2, 3, 4, 5, 6, 7, 8, 9, 10$, and three 'face cards' (King, Queen, Jack).