Define Probability and state its Axioms.

a) Probability is a numerical measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

The Axioms of Probability (Kolmogorov's Axioms) are:

1. Non-negativity: For any event $A$, the probability of $A$ is non-negative. $$P(A) \ge 0$$
2. Normalization: The probability of the sample space $S$ (the set of all possible outcomes) is 1. $$P(S) = 1$$
3. Countable Additivity: For any sequence of mutually exclusive events $A_1, A_2, A_3, \dots$ (events that cannot occur at the same time), the probability of their union is the sum of their individual probabilities. $$P(A_1 \cup A_2 \cup A_3 \cup \dots) = P(A_1) + P(A_2) + P(A_3) + \dots$$

• i) An Experiment is any process that yields an observation or an outcome. For example, tossing a coin or rolling a die.
• ii) A Sample Space ($S$) is the set of all possible outcomes of a random experiment. For example, when rolling a die, $S = \{1, 2, 3, 4, 5, 6\}$.
• iii) An Event Space (or simply an Event) is a subset of the sample space, consisting of one or more outcomes. For example, rolling an even number on a die is an event $E = \{2, 4, 6\}$.
• iv) A Sample Point is an individual outcome in the sample space. For example, when rolling a die, '3' is a sample point.
• v) Trials are individual repetitions of an experiment. For example, if you toss a coin 10 times, each toss is a trial.