Section 4.1 Randomness and 4.2 Probability Models

The Language of Probability:

• A phenomenon is random if individual outcomes are uncertain, but
  there is nonetheless a regular distribution of outcomes in several
  repetitions

• The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions
  

Thinking about Randomness:

• The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip)

Uses of Probability:

• The origins of probability were seventeenth-century games of
  chance (i.e. gambling)

• q In the eighteenth century and the nineteenth century, careful measurements in astronomy and surveying led to further advances in probability due to distributions that arise from random sampling

• Modern applications of probability apply in such diverse fields as

  • Traffic flows

  • Genetic makeups of a population

  • Energy states of subatomic particles

  • Spread of epidemics or tweets

  • Returns on risky investments

  • AI Models like ChatGPT, Claude, Gemini, etc.

Probability Models:

• The sample space S of a random phenomenon is the set of all possible outcomes

• An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

• A probability model is a description of some random phenomenon that consists of two parts: a sample space S and a probability for each outcome

Probability Rules:

• 1. Any probability is a number between 0 and 1

• 2. All possible outcomes together must have probability 1

• 3. If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities

• 4. The probability that an event does not occur is 1 minus the probabilities that the event does occur

• Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1

• Rule 2. If S is the sample space in a probability model, then P(S) = 1

• Rule 3. If A and B are disjoint, then P(A or B) = P(A) + P(B)

• This is the addition rule for disjoint events

• Rule 4: The complement of any event A is the event that A does not
  occur, written AC. P(AC) = 1 – P(A)

Assigning Probability: Finite Probability Models:

• A probability model with a finite sample space is called finite

• To assign probabilities in a finite model, list the probabilities of all the
  individual outcomes. These probabilities must be numbers between 0
  and 1 that add up to exactly 1

• The probability of an event is the sum of the probabilities of the outcomes
  making up the event
  

Assigning Probability: Equally Likely Probabilities

• a random phenomenon has k outcomes, all equally likely, then
  each individual outcome has probability 1/k. The probability of
  any event A is

• P(A) = count of outcome in A / count of outcomes in S = count of outcomes in A / k
  

Venn Diagrams:

• Sometimes, it is helpful to draw a picture to display relations among several
  events. A picture that shows the sample space S as a rectangular area and
  events as areas within S is called a Venn diagram

• Two events that are not disjoint, and the event “A and B” consisting of the outcomes they have in common

Independence and the Multiplication Rule:

• If two events A and B do not influence each other, and if knowledge about one does not change the probability of the other, the events are said to be independent of each other

• Rule 5: Multiplication Rule for Independent Events

• Two events A and B are independent if knowing that one does not change the probability that the other occurs. If A and B are independent, then

• P (A and B) = P(A) x P(B)