A pair of dice
A 'die' or dice [the plural form of die/dice is also dice] is generally made of a six faced cubic structure which can be numbered from 1 to 6 on each face. A pair of dice can have a maximum of 36 combinations or outcomes.
Probability of an event

**Basics of Probability: **Till now, we learned about exploratory analysis of data, collection of data and now we begin the transition to inference. In order to do inference, we need to use the language of probability. In order to use the language of probability, we need an understanding of random variables. In this chapter, we learn about the basic rules of probability, what it means for events to be independent, discrete and continuous random variables, simulation and rules for combining random variables.

Probability is a measure of the likeliness that an event will occur. It is used to quantify an attitude of mind towards some proposition of whose truth we are not certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty), we call probability. Thus the higher the probability of an event, the more certain we are that the event will occur. A simple example would be the toss of a fair coin. Since the 2 outcomes are deemed equiprobable, the probability of "heads" equals the probability of "tails" and each probability is 1/2 or equivalently a 50% chance of either "heads" or "tails".

**Experiment (Random phenomenon): **An experiment [i.e., random phenomenon] is an activity in which we know what outcomes could happen, but we don't know which particular outcome will happen.

**Example:** If we toss a coin, we know that we will get head (or) tail, but we don't know which particular outcome will happen. Similarly, if we roll a six sided die, we know that we will get a 1, 2, 3, 4, 5 or 6, but we don't know which particular outcome will happen.

**Trial and Outcome: ** A single attempt (or) realization of an experiment is known as trial and one of the possible results of an experiment is known as an outcome. For example: the possible outcomes for tossing a single coin is: head (or) tail; the possible outcomes for rolling a six-sided die is: 1, 2, 3, 4, 5 (or) 6.

**Sample space: **The set of all possible outcomes of an experiment is known as sample space. It is denoted by the capital letter 'S'. For example: sample space for the roll of a single die, S = {1, 2, 3, 4, 5, 6}; sample space for tossing a single coin, S = {H, T}.

**Example: **Find the sample space when two six sided dice are rolled simultaneously.

**Solution: **When two dice are rolled simultaneously, there are total 36 possible outcomes. Therefore, sample space (S) = {(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)}.

**Event: **An event is a collection of outcomes of an experiment, that is, an event is the subset of the sample space. For example: sample space for the roll of a single die, S = {1, 2, 3, 4, 5, 6}. Let event P = "face value of the die is an even number". Then P = {2, 4, 6}. Let event Q = "face value of the die is an odd number". Then Q = {1, 3, 5}. Events P and Q are subsets of the sample space.

**Occurrence of events:** In a random experiment, if E is the event of a sample space S and w is the outcome, then we say that the event E has occurred if w E. If w E, then we say that the event E has not occurred.