Probability

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.

• Probability studies the likelihood of events occurring.
• A fair coin toss has two equally likely outcomes: head or tail.
• A fair die has six equally likely outcomes: 1 to 6.
• Not all experiments have equally likely outcomes (e.g., drawing balls from a bag with different colors).
• Theoretical probability assumes equally likely outcomes.
• Theoretical probability P(E) = (Number of favorable outcomes for E) / (Total number of possible outcomes).
• 📌 Probability: Measure of likelihood of an event.
• 📌 Experiment: An action or process leading to outcomes.
• 📌 Event: A specific outcome or set of outcomes of an experiment.

Examples Illustrating Theoretical Probability

This section provides detailed examples to illustrate the calculation of theoretical probability. Example 1 considers tossing a coin once, where the event E is getting a head. Since there are two possible outcomes (head or tail) and one favorable outcome (head), the probability P(E) = 1/2. Similarly, the probability of getting a tail is also 1/2. Example 2 involves drawing a ball from a bag containing one red, one blue, and one yellow ball. Since all balls are of the same size and the draw is random, each outcome is equally likely. The probability of drawing any specific color ball is 1/3. Example 3 involves throwing a die once and finding the probability of getting a number greater than 4 (which are 5 and 6) and less than or equal to 4 (which are 1, 2, 3, 4). The probabilities are 2/6 = 1/3 and 4/6 = 2/3 respectively. These examples show how to identify the total number of possible outcomes and the number of favorable outcomes to calculate probability. The section also introduces the concept of elementary events, which are events having only one outcome, and complementary events, where the sum of the probabilities of an event and its complement is always 1.

• Probability of getting head or tail in a coin toss is 1/2 each.
• Probability of drawing any one colored ball from equally likely options is 1/3.
• Probability of getting a number greater than 4 on a die is 1/3.
• Elementary event: Event with only one outcome.
• Sum of probabilities of all elementary events is 1.
• Complementary events: P(E) + P(not E) = 1.
• 📌 Elementary event: An event with exactly one outcome.
• 📌 Complement of an event (¬E): Event that E does not occur.
• 📌 Complementary events: Two events whose probabilities sum to 1.