What is Probability? State the Axioms of Probability. Write shote notes on the following: i) Experiment ii) Sample Space iii) Event Space iv) Sample Point v) Trials. If a fair die is tossed once, what is the probability that a perfect square did not show up?

The probability of any event $A$ is a non-negative real number.

$$P(A) \ge 0$$

The probability of the sample space $S$ (the set of all possible outcomes) is 1.

$$P(S) = 1$$

For any sequence of mutually exclusive events $A_1, A_2, A_3, \dots$, the probability of their union is the sum of their individual probabilities.

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

1. b) Write short notes on the following:

i) Experiment
An experiment is any process that generates a set of well-defined outcomes. Examples include tossing a coin, rolling a die, or measuring a person's height.

ii) Sample Space
The sample space ($S$) of an experiment is the set of all possible outcomes. For example, when rolling a standard six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$.

iii) Event Space
An event space (or simply an event) is a subset of the sample space. It represents a specific outcome or a collection of outcomes. For example, when rolling a die, the event "rolling an even number" is the subset $\{2, 4, 6\}$.

iv) Sample Point
A sample point is an individual outcome in the sample space. It is a single element of the sample space. For example, when rolling a die, '3' is a sample point.

v) Trials
A trial is a single performance of an experiment. If an experiment is repeated multiple times, each repetition is considered a trial. For example, tossing a coin 10 times involves 10 trials.