Probability basics

Probability Basics

Defining Probability

  • Probability is the measure of the likelihood that an event will occur in a random experiment.
  • It is quantified as a number between 0 and 1.
  • An event is a specific result or a set of results of a random experiment or process.
  • An event with probability 0 is considered impossible, and one with probability 1 is considered certain.

Understanding Events

  • Independent events: Events that are not affected by each other. The occurrence of one event does not change the probability of the other event.
  • Dependent events: The probability of one event depends on the outcome of another. The occurrence of one event does change the probability of the other event.
  • Mutually exclusive events: Two events that cannot both happen at the same time. If one happens, the other cannot.

Fundamental Concepts

  • Sample Space: This is the set of all possible outcomes of a random experiment. It is typically denoted by the Greek letter Omega (Ω).
  • Event Space: A subset of the sample space. Any subset of sample space, including the empty set and the sample space itself.
  • Favourable Outcomes: These are the outcomes of an experiment which are in line with what we are looking for.

Calculating Probability

  • The probability (P) of an event A is calculated as: P(A) = Number of favourable outcomes / Total number of outcomes.
  • For mutually exclusive events, probabilities can be added together.
  • For independent events, probabilities should be multiplied.
  • The Complementary Rule states that the sum of the probability of an event and its complement is equal to 1. So, P(A) + P(A’) = 1, where A’ is the complement of the event A.

Conditional Probability

  • Conditional probability measures the probability of an event given that (by condition) another event has occurred.
  • It is denoted as P(A B), which means the probability of A given that B has occurred.
  • For dependent events, the conditional probability is calculated as: P(A B) = P(A ∩ B) / P(B), where ∩ denotes intersection.

Interpretation of Probability

  • Interpretation of probability often confuses beginners. There are two school of thoughts: frequency interpretation (or empirical interpretation) and subjective interpretation.
  • In frequency interpretation, probabilities are considered as long term frequency of events (E.g: If an unbiased coin is flipped many times and most of the time it shows up head.)
  • In subjective interpretation, people’s degree of belief in an event is considered as probability.