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


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

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%).

Basic Probability Formulas

Probability Range

0 ≤ P(A) ≤ 1

Rule of Complementary Events

P(AC) + 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

FX(x) = P(Xx)

Probability Mass Function

sum(i=1..n, P(X=x(i)) = 1

Probability Density Function

fX(x) = dFX(x)/dx

FX(x) = integral(-inf..x, fX(y)*dy)

FX(x) = sum(k=1..x, P(X=k))

P(a<=X<=b) = integral(a..b, fX(x)*dx)

integral(-inf..inf, fX(x)*dx) = 1


Cox(X,Y) = E(X-ux)(Y-uy) = E(XY) - ux*uy


corr(X,Y) = Cov(X,Y)/(Std(X)*Std(Y))

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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