Definition of Probability
Probability in mathematics is a measure of the likelihood that a particular event will occur, expressed as a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. It is formally defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space, assuming all outcomes are equally likely.
Key Principles of Probability
Fundamental principles include the addition rule for mutually exclusive events, which states that the probability of A or B occurring is P(A) + P(B), and the multiplication rule for independent events, where P(A and B) = P(A) × P(B). Probability also adheres to axioms such as non-negativity, normalization (total probability sums to 1), and additivity for disjoint events.
Practical Example
Consider flipping a fair coin: the probability of getting heads is 1/2, as there is one favorable outcome (heads) out of two possible outcomes (heads or tails). For rolling a six-sided die, the probability of rolling an even number (2, 4, or 6) is 3/6 = 1/2, illustrating how probabilities are calculated from equally likely outcomes.
Importance and Applications
Probability is essential in fields like statistics, finance, and science for modeling uncertainty, predicting outcomes, and informing decision-making. It underpins risk assessment in insurance, hypothesis testing in research, and algorithms in machine learning, enabling reliable analysis of random phenomena in the real world.