MATHEMATICS CLASS NOTES: PROBABILITY FOR SSS 2

Theoretical Frameworks

To master the study of probability, we must first understand the mathematical models and philosophies that govern how we measure uncertainty. Probability is the branch of mathematics that quantifies the likelihood of an event occurring. Historically, this study evolved from games of chance in the 17th century, championed by mathematicians such as Blaise Pascal and Pierre de Fermat. Today, it forms the bedrock of statistics, insurance, quantum mechanics, and artificial intelligence.

In modern mathematics, we analyze probability through three primary theoretical frameworks:

1. The Classical (A Priori) Approach

This framework is applied when all possible outcomes of an experiment are known, finite, and equally likely to occur. It does not require physical experimentation; instead, it relies on deductive reasoning.

Mathematically, if an experiment can result in $N$ mutually exclusive, equally likely outcomes, and if $n$ of these outcomes are favorable to an event $A$ , then the classical probability of event $A$ , denoted by $P(A)$ , is defined as:

$P(A) = \frac{n(A)}{n(S)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Limitation: This approach fails when the total number of outcomes is infinite, or when the outcomes are not equally likely (for example, a biased coin or a weighted die).

2. The Empirical (Frequentist/Experimental) Approach

When outcomes are not equally likely, or when the sample space is too complex to determine theoretically, we rely on observation and historical data. The empirical probability of an event is the limit of its relative frequency as the number of trials increases indefinitely. This is governed by the Law of Large Numbers.

If an experiment is performed $N$ times under identical conditions, and event $A$ occurs $f$ times, the empirical probability of $A$ is:

$P(A) = \lim_{N \to \infty} \frac{f}{N}$

In practical terms, for a sufficiently large number of trials $N$ :

$P(A) \approx \frac{\text{Frequency of event } A}{\text{Total number of trials } N}$

Example: If a factory tests 10,000 lightbulbs and finds 150 defective ones, the empirical probability of a bulb being defective is $\frac{150}{10000} = 0.015$ or $1.5\%$ .

3. The Axiomatic Approach

Developed by Andrey Kolmogorov in 1933, this is the modern, mathematically rigorous foundation of probability. It uses set theory to define probability without relying on physical experiments or the assumption of equally likely outcomes. The axiomatic framework states that probability is a real-valued function $P$ that assigns a real number $P(A)$ to every event $A$ in a sample space $S$ , satisfying three core axioms:

Axiom of Non-negativity: For any event $A$ , $P(A) \geq 0$ . Probability cannot be negative.
Axiom of Certainty (Normalization): The probability of the entire sample space $S$ is $1$ , i.e., $P(S) = 1$ .
Axiom of Additivity: For any sequence of mutually exclusive events $A_1, A_2, A_3, \dots$ (where $A_i \cap A_j = \emptyset$ for $i \neq j$ ):

$P(A_1 \cup A_2 \cup A_3 \dots) = P(A_1) + P(A_2) + P(A_3) + \dots$

All mathematical properties and formulas of probability are logically derived from these three basic axioms.

In-Depth Core Concepts

To build a solid foundation, we must define and analyze the core building blocks of probability.

Experiment or Trial

An experiment (or trial) is any systematic, repeatable process or action that generates a well-defined set of observable results. The key characteristic of a random experiment is that while the set of all possible results is known beforehand, the exact outcome of any single trial cannot be predicted with absolute certainty.

Example 1: Rolling a standard six-sided die is a random experiment. We know the possible results are numbers from 1 to 6, but we cannot predict which number will land face-up on the next roll.
Example 2: Drawing a single card from a well-shuffled deck of 52 playing cards.

Outcome

An outcome is a single, specific, and indivisible result obtained from running a single trial of an experiment.

Example: If you toss a coin once, "Heads" ( $H$ ) is one outcome, and "Tails" ( $T$ ) is another outcome. If you roll a die, getting a " $4$ " is a single outcome.

Sample Space ( $S$ )

The sample space (denoted by $S$ or $\Omega$ ) is the set containing all possible outcomes of a random experiment. It represents the universal set for that specific experiment. Defining the sample space accurately is the first and most critical step in solving any probability problem.

Let us examine the sample spaces of common random experiments:

A. Tossing Coins

One Coin: When a single coin is tossed, there are only two possible outcomes: Heads ( $H$ ) or Tails ( $T$ ). $S = \{H, T\}, \quad n(S) = 2$
Two Coins: When two coins are tossed simultaneously (or one coin is tossed twice), the sample space consists of ordered pairs, representing the outcome of the first coin and the second coin. $S = \{(H,H), (H,T), (T,H), (T,T)\}, \quad n(S) = 4$
Three Coins: $S = \{(H,H,H), (H,H,T), (H,T,H), (H,T,T), (T,H,H), (T,H,T), (T,T,H), (T,T,T)\}, \quad n(S) = 8$ Note: The general rule for tossing $k$ coins is $n(S) = 2^k$ .

B. Rolling Dice

One Die: A standard die has six faces numbered 1 to 6. $S = \{1, 2, 3, 4, 5, 6\}, \quad n(S) = 6$
Two Dice: When two distinct dice (e.g., a red die and a blue die) are rolled together, the sample space consists of ordered pairs $(x, y)$ , where $x$ is the score on the first die and $y$ is the score on the second die. This can be represented as a $6 \times 6$ grid:

(1,1) & (1,2) & (1,3) & (1,4) & (1,5) & (1,6) \\ (2,1) & (2,2) & (2,3) & (2,4) & (2,5) & (2,6) \\ (3,1) & (3,2) & (3,3) & (3,4) & (3,5) & (3,6) \\ (4,1) & (4,2) & (4,3) & (4,4) & (4,5) & (4,6) \\ (5,1) & (5,2) & (5,3) & (5,4) & (5,5) & (5,6) \\ (6,1) & (6,2) & (6,3) & (6,4) & (6,5) & (6,6) \end{array}$$ $$n(S) = 6 \times 6 = 36$$

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