Basic probability

Basic Probability Definition

What is probability theory? A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

Fundamental Terms

Random Experiment = A random experiment is a physical situation whose outcome cannot be predicted until it is observed.

Sample Space = A sample space, is a set of all possible outcomes of a random experiment.

Random Variables = A random variable, is a variable whose possible values are numerical outcomes of a random experiment. There are two types of random variables.

Discrete Random Variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,…….. Discrete random variables are usually (but not necessarily) counts.

Continuous Random Variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements.

Probability = Probability is the measure of the likelihood that an event will occur in a Random Experiment. Probability is quantified as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes (“heads” and “tails”) are both equally probable; the probability of “heads” equals the probability of “tails”; and since no other outcomes are possible, the probability of either “heads” or “tails” is 1/2 (which could also be written as 0.5 or 50%).

Conditional Probability = Conditional Probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, is usually written as P(A|B).These are the most fundamental terms according to probability. Now let's have a look at some Formulas.

Basic Probability Formulas

Probability Range

0 ≤ P(A) ≤ 1

Rule of Complementary Events

P(A^C) + P(A) = 1

Rule of Addition

P(A∪B) = P(A) + P(B) - P(A∩B)

Disjoint Events

Events A and B are disjoint iff

P(A∩B) = 0

Conditional Probability

P(A | B) = P(A∩B) / P(B)

Bayes Formula

P(A | B) = P(B | A) ⋅ P(A) / P(B)

Independent Events

Events A and B are independent iff

P(A∩B) = P(A) ⋅ P(B)

Cumulative Distribution Function

F_X(x) = P(X ≤ x)

Probability Mass Function

Probability Density Function

Covariance

Correlation

Bernoulli: 0-failure 1-success

Geometric: 0-failure 1-success

Hypergeometric: N objects with K success objects, n objects are taken.

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