Chapter 4 Probability and Counting Rules

Introduction “The only two sure things are death and taxes” A cynical person once said

Introduction Probability – The chance of an event occurring.

Introduction Probability – The chance of an event occurring. Probability is the basis for inferential statistics.

Sample Spaces and Probability Chapter 4 Section 1

Sample Spaces and Probability Flipping a coin, rolling a die, or drawing a card are all called probability experiments.

Sample Spaces and Probability Probability Experiment – A probability experiment is a chance process that leads to well-defined results called outcomes. Outcome – An outcome is the result of a single trial of a probability experiment.

Sample Spaces and Probability Probability Experiment – A probability experiment is a chance process that leads to well-defined results called outcomes. Outcome – An outcome is the result of a single trial of a probability experiment. A trial means flipping a coin once, roling one die once, or the like.

Sample Spaces and Probability Sample Space – A sample space is the set of all possible outcomes of a probability experiment. Examples

Sample Spaces and Probability Event – An event consists of a set of outcomes of a probability experiment. Simple Event – An event with one outcome is called a simple event. Compound Event – An event with more than one outcome is called a compound event.

Sample Spaces and Probability Tree Diagrams – A tree diagram is a device of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment.

Sample Spaces and Probability Three basic interpretations of probability 1.Classical Probability 2.Empirical or Relative Frequency Probability 3.Subjective Probability

Classical Probability Classical probability uses sample spaces to determine the numerical probability that an event will happen. Classical probability assumes that all outcomes in the sample space are equally likely to occur.

Classical Probability

Rounding Rule for Probabilities 1.Reduce all fraction to lowest form. 2.Rounded to two or three decimals. 3.If number is really small it is permissible to round to the first nonzero digit after the point.

What does “and” and “or” mean In probability “and” means at the same time. In probability “or” has two cases: 1.Inclusive 2.Exclusive

Probability Rules

3.If an event E is certain, then the probability of E is 1. 4.The sum of all probabilities of all the outcomes in the sample space is 1.

Complementary Events

Classical Probability Venn Diagrams – One pictorial representation of probabilities. – Read from your book the two paragraphs on Venn diagrams (pg ).

Empirical Probability – Empirical probability relies on actual experience to determine the likelihood of outcomes. – Difference between classical and empirical probabilities is that classical assumes certain outcomes are equally likely.

Formula for Empirical Probability

Empirical Probabilities Examples

Empirical Probability Empirical probabilities can also be found by using a relative frequency distribution. – The frequencies are the same as the relative frequencies.

Law of Large Numbers The Law of large numbers claims that if an experiment is conducted enough then the Classical probability will be approximately the same as the empirical probability.

Subjective Probability – Subjective probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information. – This guess is based on the person’s experience and evaluation of a situation.