PSDRV- Probability and Special Discrete Random Variables Overview

Math_APStatBanner.png Probability and Special Discrete Random Variables

Introduction

Chance is all around us. Probability is the branch of mathematics that describes the pattern of chance outcomes. The reasoning of statistical inference rests on asking, "How often would this method give a correct answer if I used it very many times?"   When we produce data by random sampling or randomized comparative experiments the laws of probability answer that question. The fundamental concepts of probability are the basis for inference. The tools you acquire will eventually help you describe the "behavior" of statistics from random samples and randomized comparative experiments.
The term random in statistics is NOT a synonym for "haphazard" but a description of a kind of "order or pattern" that emerges after many trials or repetitions.   The idea of probability is empirical which means based on observation rather than theorizing.   Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run.   You cannot predict the outcome of tossing "a" coin one time but if you make many, many tosses a regular pattern will emerge...this is the REMARKABLE fact that forms the basis for probability.

Essential Questions

  • Is anything really random?
  • Do chance occurrences demonstrate any patterns over time?
  • Are there rules that govern random events?
  • How do we use probability to describe the likelihood of an event occurring?
  • Are there any "special" models under the probability umbrella?
  • What is a random variable?  

Key Terms

The following key terms will help you understand the content in this module.

Probability and Special Discrete Random Variables Key Terms

probability - 0< P(A)< 1 is a decimal number that reports the likelihood of an event's occurrence using any capital letter to name the event

Random- values of individual outcomes of a phenomenon are uncertain but a regular distribution evolves over a large number of repetitions

Event- collection of outcomes usually identified so that a probability may be attached

P(A) - notation for the probability of the occurrence of event

A sample space- collection of all possible outcome values with total probability = 1

Independence - knowledge of one event occurring does not influence the probability of the other event occurring - symbolically two events are independent if P(A) = P(A|B)

Disjoint- 2 or more events that share no outcomes…do not overlap…indicated with Venn diagram

Venn diagram- model used to help find probabilities for union or intersection of 2 or more collections (sets)

Union (OR operator)- union of any collection of events is the event that at least one of the collection occurs

Intersection (AND operator)- intersection of any collection of events that occur simultaneously

Tree diagram-branch display of conditional events or probabilities

Complement - probability of event occurring is (1 - probability that it doesn't occur) and vice versa

Replacement- probability of subsequent selection of objects from a collection remain exactly alike when selected objects are returned to the collection

Joint probability-the probability of simultaneous occurrence of events conditional

Probability- probability of a second event knowing that a first event has already occurred

 

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