Science: Life Lives On

Probability Terminology

• An experiment/trial is any process generating well-defined outcomes.
• Sample space refers to the set of all possible outcomes.
• An event is a subset of the sample space.

Approaches to Assigning Probability

• There are three approaches to assigning probability: Classical, Relative Frequency, and Subjective.

Classical Approach (Theoretical Probability)

• Applies when all outcomes are equally likely.
• Probability of an event E, P(E), equals outcomes favorable to E divided by total outcomes.

Relative Frequency Approach (Empirical Probability)

• Based on historical data or experiments.
• Probability of an event E, P(E), equals number of times E occurs divided by the total trials.

Subjective Approach

• Based on personal judgment, belief, or expertise.
• Useful when historical data is unavailable, or events are unique.

Rules of Probability

• Probability values range from 0 to 1: 0 ≤ P(E) ≤ 1.
• The sum of probabilities for all possible outcomes equals 1: Σ P(Ei) = 1.
• The probability of an empty set is 0: P(∅) = 0.

Complement Rule

• The complement of E (E' or E^c) contains all outcomes not in E.
• P(E) + P(E') = 1, therefore P(E') = 1 - P(E).

Addition Rule

• For events A and B, P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• If A and B are mutually exclusive (disjoint), then P(A ∩ B) = 0 and so P(A ∪ B) = P(A) + P(B).

Conditional Probability

• Probability of event A, given event B occurred, is denoted as P(A|B).
• P(A|B) = P(A ∩ B) / P(B), given that P(B) > 0.

Multiplication Rule

• $P(A \cap B) = P(A|B) \cdot P(B)$, or $P(A \cap B) = P(B|A) \cdot P(A)$.

Independence

• Events A and B are independent if one's occurrence doesn't affect the other's probability.
• Test includes: P(A|B) = P(A) or P(B|A) = P(B) or P(A ∩ B) = P(A) * P(B).

Discrete Probability Distributions

• Discrete variables can only take finite or countably infinite values.

Bernoulli Distribution

• Models a single trial with success (p) or failure (1-p).
  • P(X = x) = p^x (1 - p)^(1-x), for x ∈ {0, 1}
• E[X] = p
• Var(X) = p(1 - p)

Binomial Distribution

• Models successes in fixed independent Bernoulli trials.
  • P(X = x) = (n choose x) * p^x * (1 - p)^(n-x), for x ∈ {0, 1,..., n}
  • Where (n choose x) = n! / (x!(n - x)!)
• E[X] = np
• Var(X) = np(1 - p)

Poisson Distribution

• Models events occurring in a fixed interval.
  • P(X = x) = (e^(-λ) * λ^x) / x!, for x ∈ {0, 1, 2,...}
• E[X] = λ
• Var(X) = λ
• X is the random variable.
• x is a specific X value.
• P(X = x) is the probability that X takes value x.
• E[X] is X's expected value (mean).
• Var(X) is X's variance.
• n is the number of trials.
• p is a single trial's success probability.
• λ (lambda) is the average rate of events.

Matrices Definition

• A matrix A of type m × n is a table of m.n elements in m rows and n columns.
• $A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... &... &... &... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$
  • m is the number of rows
  • n is the number of columns
  • $a_{ij}$ represents the element's location in row i and column j

Example Matrix

• $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$
• Is a matrix of 2x3, with 2 rows and 3 columns.

Types of Matrices

Square Matrix

• Has the same number of rows and columns, or m = n.
  • $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$

Row Matrix

• Has only 1 row, or m = 1.
  • $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

Column Matrix

• Has only 1 column, or n = 1.
  • $A = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$

Null Matrix

• All elements are zero.
  • $A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$

Identity Matrix

• It's a square matrix in which all elements of the main diagonal are 1 and the remaining elements are 0.
  • $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

Transpose Matrix

• Given matrix A, its transpose, is shown as $A^T$, and it's obtained by swapping the rows with the columns of A.
  • If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$

Matrix Operations

Matrix Addition

• The sum of 2 matrices A and B of the same type, or m × n, is obtained by the sum of the corresponding elements.
  • If $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}$, then $A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}$

Matrix Multiplication by Scalar

• The multiplication of a matrix A by a scaler k, is obtained by multiplying each element of the matrix by k.
  • If $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$, then $kA = \begin{bmatrix} ka_{11} & ka_{12} \ ka_{21} & ka_{22} \end{bmatrix}$

Matrix Multiplication

• Multiplication of two matrices A (m × n) and B (n × p) is written as:
• The element $c_{ij}$ of the resulting matrix C is found by adding the products of the elements in row i of A by the elements of column j of B.
  • $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}$, then $AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}$

Determinant

• It's a function that associates with each square matrix a scaler.
• The determinant of a matrix A is denoted det(A) or |A|.
• Calculated for 2x2 matrices:
  • If $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, then $det(A) = ad - bc$
• The determinant for a 3x3 matrix can be calculated using the Sarrus rule.

Matrix Inverse

• A matrix a is inversible if a matrix $A^{-1}$ exists where $AA^{-1} = A^{-1}A = I$, where I is the identity matrix.
• The inverse of a matrix A only exists if the determinent of A is not zero.
• The inverse of a 2x2 matrix is calculated with:
  • If $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, then $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$