Probability: Sample Space and Events

Introduction to Probability

• An experiment is an activity that has an observable outcome.
  • Examples include tossing a coin, drawing a card from a deck, or rolling a die.
• Sample space is the set of all possible outcomes of an experiment.
  • Coin toss sample space : $S = {H, T}$
  • Sample space for drawing a card from a deck: $S = {A\clubsuit, K\clubsuit,..., 2\clubsuit, A\diamondsuit, K\diamondsuit,..., 2\diamondsuit, A\heartsuit, K\heartsuit,..., 2\heartsuit, A\spadesuit, K\spadesuit,..., 2\spadesuit}$
  • Die roll sample space: $S = {1, 2, 3, 4, 5, 6}$
• An event is a subset of the sample space.
  • An event in a coin toss could be: observing a Head, $E = {H}$
  • An event in card drawing could be: observing an Ace, $E = {A\clubsuit, A\diamondsuit, A\heartsuit, A\spadesuit}$
  • An event in die rolling could be: observing an even number, $E = {2, 4, 6}$

Axioms of Probability

• The probability of any event is a real number between 0 and 1: $0 \le P(E) \le 1$
• The probability of the sample space is 1, $P(S) = 1$
• For mutually exclusive events $E_1, E_2, E_3,...$, $P(E_1 \cup E_2 \cup E_3 \cup...) = P(E_1) + P(E_2) + P(E_3) +...$

Rules of Probability

• The probability of the null set is 0: $P(\emptyset) = 0$
• For events E and F, $P(E \cup F) = P(E) + P(F) - P(E \cap F)$
• For mutually exclusive events E and F, $P(E \cup F) = P(E) + P(F)$
• If E and $E^c$ are complementary events then: $P(E^c) = 1 - P(E)$

Discrete Probability

• For a discrete random variable X, Probability Mass Function (PMF) provides the probability that X takes a specific value x: $p(x) = P(X = x)$

• Properties of PMF:

  • $0 \le p(x) \le 1$
  • $\sum_{x}^{} p(x) = 1$

• For a discrete random variable X, Cumulative Distribution Function (CDF) provides the probability that X takes on a value less than or equal to x.

  $F(x) = P(X \le x) = \sum_{u \le x}^{}p(u)$

• Properties of CDF:

  • $0 \le F(x) \le 1$
  • $F(x)$ is a non-decreasing function

Continuous Probability

• For a continuous random variable X, Probability Density Function (PDF) is a function such that:
  • $P(a \le X \le b) = \int_{a}^{b} f(x)dx$
• The properties for a PDF ( Probability Density Function ):
  • $f(x) \ge 0$
  • $\int_{-\infty}^{\infty} f(x)dx = 1$
• For a continuous random variable X, Cumulative Distribution Function (CDF) is defined as
  • $F(x) = P(X \le x) = \int_{-\infty}^{x}f(u)du$
• Properties of CDF:
  • $0 \le F(x) \le 1$
  • $F(x)$ is a non-decreasing function

Vectores (Vectors - Spanish)

• A Vector is a line segment with orientation that has magnitude, direction and orientation.
• A Vector $\vec{A}$ is defined by its components $A_x$ and $A_y$: $\vec{A} = (A_x, A_y)$
  • $A_x = A \cos \theta$
  • $A_y = A \sin \theta$
  • $A = \sqrt{A_x^2 + A_y^2}$
  • $\theta = \tan^{-1} \left( \frac{A_y}{A_x} \right)$

Sum of Vectors

• Given two vectors $\vec{A} = (A_x, A_y)$ and $\vec{B} = (B_x, B_y)$, their sum is: $\vec{A} + \vec{B} = (A_x + B_x, A_y + B_y)$

Dot Product

• Given two vectors $\vec{A}$ and $\vec{B}$, their dot product is $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta = A_x B_x + A_y B_y + A_z B_z$
• $\theta$ is the angle between $\vec{A}$ and $\vec{B}$

Cross Product

• For two vectors $\vec{A}$ and $\vec{B}$, their cross product results in a vector $\vec{C}$ defined as $\vec{C} = \vec{A} \times \vec{B}$.
  • Magnitude: $|\vec{C}| = |\vec{A}| |\vec{B}| \sin \theta$
  • Direction: Perpendicular to the plane formed by $\vec{A}$ and $\vec{B}$.
  • Sense: According to the right-hand rule.
  • Components:
    • $\vec{A} \times \vec{B} = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x)$
    • $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$
    • $\hat{i}$, $\hat{j}$ and $\hat{k}$ are the unit vectors in the directions x, y, and z, respectively.