8/23/2017

In Lecture 1

  • Potential Outcome: One specific result from a random experiment

  • Element \(\omega\): a single potential outcome

  • Event: a collection of some outcomes
    • empty set \(\emptyset\): no outcomes
    • sample space \(S\) or \(\Omega\): consists of all outcomes
  • Union \(\cup\): a new set that contains each outcome found in an of the component events
  • Intersection \(\cap\): a new set that contains only the outcomes found in all of the component event.

Probability Rules

Disjoint

  • A pair of events A, B is disjoint, if they have no outcome in common

  • \(A\cap B = \emptyset\)

  • Also called mutually exclusive

  • A collection of event is disjoint if every pair of events is disjoint

Three Probability Axioms

  1. For each event A, \[0\le P(A) \le 1\]
  2. For the sample space S, \[P(S) = 1\]
  3. If \(A_1, A_2, \cdots\) is a collection of disjoint events, then \[P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^\infty P(A_i)\]

Two Thoerem Based on Probability Axioms

  1. The probability of the empty set \(\emptyset\) is always 0

  2. If \(A_1, A_2, \cdots, A_n\) is a collection of finitely many disjoint events, then the probability of the union of the events equals the sum of the probabilities of the events: \[P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i)\]

Equally Likely Events

  • Revisit on rolling a die

  • Theorem: If s sample space S has n equally likely outcomes, then each outcome has probability \(\frac{1}{n}\) of occuring

  • Corollary: If sample space S has n equally likely outcomes, and A is an event with j outcomes, then event A has probability \(\frac{j}{n}\) of occuring, i.e., \[P(A) = \frac{j}{n}\]

  • Corollary If sample space S has a finite number of equally likely outcomes, then event A has probability \[P(A) = \frac{|A|}{|S|}\]

Probability with Several Events

DeMorgan's law

  1. \[(A\cup B)^c = A^c \cap B^c \]

  2. \[(A\cap B)^c = A^c \cup B^c \]

General addition rule

  • \[P(A\cup B) = P(A) + P(B) - P(A\cap B) \]

Example 1

Three of the major commercial computer operating systems are Windows, Mac OS, and Red Hat Linux Enterprise. A Computer Science professor selects 50 of her students and asks which of these three operating systems they use. The results for the 50 students are summarized below.

  • 30 students use Windows
  • 16 students use at least two of the operating systems
  • 9 students use all three operating systems
  • 18 students use Mac OS
  • 46 students use at least one of the operating systems
  • 11 students use both Windows and Linux
  • 11 students use both Windows and Mac OS

Example 1 continued

Let Windows = W, Mac OS = M, and Red Hat Linux Enterprise = L

  1. \(N(W^c \cap M^c)\)
  2. \(P(W^c \cup M^c)\)
  3. \(N(W \cup M \cup L)\)

Example 2

In a certain population, 10% of the population are rich, 5% are famous, and 3% are both.

  • What is the probability a randomly chosen person is not rich?
  • What is the probability a randomly chosen person is rich but not famous?
  • What is the probability a randomly chosen person is either rich or famous?
  • What is the probability a randomly chosen person is either rich or famous but not both?
  • What is the probability a randomly chosen person has neither wealth nor fame?