MP Board · Class 10 · Mathematics · ProbabilityExplain the concept of theoretical (classical) probability in detail. Discuss its fundamental definitions, properties, sample space, elementary events, and the relationship between the probability of an event and its complementary event with proper mathematical expressions and headings.

Introduction to Probability\nProbability is a branch of mathematics that deals with the numerical evaluation of the likelihood of the occurrence of an event. In Class 10 Mathematics, we primarily deal with Theoretical Probability (also called classical probability), which is based on equally likely outcomes of a random experiment.

1. Random Experiment and Sample Space

• Random Experiment: An experiment whose outcome cannot be predicted with certainty in advance, even though all possible outcomes are known, is called a random experiment. For example, tossing a coin or rolling a die.
• Sample Space ($S$): The set of all possible outcomes of a random experiment is called the sample space. For instance, when tossing a single die, the sample space is $S = {1, 2, 3, 4, 5, 6}$.

2. Elementary Events and Events

• Elementary Event: An outcome of a random experiment is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is always 1.
• Event ($E$): A subset of the sample space is called an event. An event can contain one or more elementary events. If an event contains only one element, it is called an elementary event.

3. Definition of Theoretical Probability\nFor a random experiment, if we assume that the outcomes are equally likely, the theoretical probability of an event $E$, denoted by $P(E)$, is defined as:

$$P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of possible outcomes of the experiment}} = \frac{n(E)}{n(S)}$$\nWhere $n(E)$ is the number of elements in event $E$, and $n(S)$ is the total number of elements in the sample space $S$.

4. Key Properties of Probability

• Range of Probability: The probability of any event $E$ always lies between 0 and 1 (inclusive). Mathematically, $0 \le P(E) \le 1$.
• Sure Event (Certain Event): An event that is certain to occur has a probability of 1. It is denoted as $P(S) = 1$.
• Impossible Event: An event that cannot possibly occur has a probability of 0. It is denoted as $P(\phi) = 0$.

5. Complementary Events

• For any event $E$, the event "not $E$" represents the complementary event of $E$, denoted by $\bar{E}$ or $E'$.
• The sum of the probability of an event and its complementary event is always equal to 1: $$P(E) + P(\bar{E}) = 1$$ This fundamental relationship allows us to easily find the probability of the non-occurrence of an event if its occurrence probability is known, expressed as $P(\bar{E}) = 1 - P(E)$.