Probability | Class 10 Mathematics Notes

14.1 Probability — A Theoretical Approach

Probability is a branch of mathematics that deals with the study of chance or likelihood of occurrence of events. It helps us predict how likely an event is to happen. The theoretical approach to probability assumes that the outcomes of an experiment are equally likely. For example, when a fair coin is tossed, it can land either head up or tail up, and both outcomes are equally likely because the coin is unbiased and symmetrical. Similarly, when a fair die is rolled, the possible outcomes are 1, 2, 3, 4, 5, and 6, each equally likely. However, not all experiments have equally likely outcomes. For instance, if a bag contains 4 red balls and 1 blue ball, the probability of drawing a red ball is higher than drawing a blue ball, so the outcomes are not equally likely. Despite this, in this chapter, we assume all experiments have equally likely outcomes to simplify the study of theoretical probability. The theoretical probability of an event E, denoted as P(E), is defined as the ratio of the number of outcomes favorable to E to the total number of possible outcomes of the experiment, assuming all outcomes are equally likely. This definition was given by Pierre Simon Laplace in 1795. The empirical or experimental probability, introduced in Class IX, is based on actual trials and is the ratio of the number of trials in which the event happened to the total number of trials. However, empirical probability may not always be feasible, especially for events that are expensive or impossible to repeat many times, such as satellite launches or earthquakes. The theoretical probability provides a way to calculate exact probabilities based on assumptions rather than repeated trials.

📊 Diagram: See figure_1: Pierre Simon Laplace (1749 – 1827)

🧪 Activity: Consider tossing a coin or rolling a die to observe equally likely outcomes and calculate theoretical probability.

🔗 Connection: This section lays the foundation for understanding probability by defining key concepts and the theoretical approach, leading to detailed examples and applications in subsequent sections.