Probability theory
This presentation is the property of its rightful owner.
Sponsored Links
1 / 34

Probability theory PowerPoint PPT Presentation


  • 35 Views
  • Uploaded on
  • Presentation posted in: General

Probability theory. Chapter 14 sec 1. Movie Quotes.

Download Presentation

Probability theory

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Probability theory

Probability theory

Chapter 14 sec 1


Movie quotes

Movie Quotes

  • "In this galaxy, there's a mathematical probability of three million Earth type planets.  And in all of the universe three million, million galaxies like this.  And in all of that, and perhaps more, only one of each of us.  Don't destroy the one named 'Kirk.'"


Probability theory

  • Kirk: Mr. Spock, have you accounted for the variable mass of whales and water in your time re-entry program? Spock: Mr. Scott cannot give me exact figures, Admiral, so... I will make a guess. Kirk: A guess? You, Spock? That's extraordinary.


Famous quotes

Famous quotes

  • AristotleThe probable is what usually happens.

  • Bertrand, JosephCalcul des probabilitésHow dare we speak of the laws of chance? Is not chance the antithesis of all law?


Probability theory

  • Joseph Louis François Bertrand (March 11, 1822 – April 5, 1900, born and died in Paris) was a Frenchmathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics.


Probability theory

  • Caesar, JuliusIactaalea est. (The die is cast.)

  • Doyle, Sir Arthur ConanThe Sign of FourWhen you have eliminated the impossible, what ever remains, however improbable, must be the truth.


Random phenomena

Random phenomena

  • What is random phenomena?

    • Occurrences that vary from day to day and case to case.

    • Weather conditions, rolling dice at craps or Monopoly, drilling oil, driving your car.

    • We never know exactly how a random phenomena will turn out, we often can calculate a number called probability.


Experiment

Experiment

  • Def.

    • Is any observation of a random phenomenon.


Outcome

Outcome

  • Def.

    • The different possible results of the experiment


Sample space

Sample Space

  • Def.

    • The set of all possible outcomes for an experiment.


Finding the sample space example 1

Finding the sample space.Example 1

  • We select an iPhone from a production line and determine whether it is defective.

  • The sample space is;

    • {defective, nondefective}


Example 2

Example 2

  • Three children are born to a family and we note the birth order with respect to gender.

  • Make a tree diagram and find all the possibilities.

    • {bbb,bbg,bgg,bgb,gbb,gbg, ggb,ggg}


Event

Event

  • Def.

    • In probability theory, an event is a subset of the sample space.


Write each event as a subset of the sample space

Write each event as a subset of the sample space.

  • A tails occurs when we flip a single coin.

  • {Tails}

  • Two girls and one boy are born in a family.

  • {ggb,gbg,bgg}


Probability of an outcome

Probability of an outcome

  • Def.

    • In a sample space is an number between 0 and 1 inclusive. The sum of the probabilities of all the outcomes in the sample space must be 1.


Probability theory

Certain

1.0

Likely to occur

0.5

50-50 Chance of occurring

Not likely to occur

Impossible

0.0


Probability of an event e

Probability of an event (E)

  • Def.

    • P(E) is defined as the sum of the probabilities of the outcome that make up E.


Probability theory

  • One way to determine probabilities is to use empirical information. Meaning we make observations and assign probabilities based on those observations.


Empirical assignment of probabilities

Empirical assignment of Probabilities

  • If E is an event and we perform an experiment several times, then we estimate the probability of E as follows;


Formula

Formula


Probability theory

  • The residents of a small town and the surrounding area are divided over the proposed construction of a spring car racetrack in the town.


Table

Table


Problem

Problem

  • If a newspaper reporter randomly selects a person to interview from these people,

    • What is the probability that the person supports the racetrack?


Probability theory

  • What are the odds in favor of the person supporting the racetrack?

  • We normally say, 5 to 4


Cal probability when outcomes are equally likely

Cal. Probability when outcomes are equally likely.

  • If E is an event in a sample space S with all equally likely outcomes, then the probability of E is given by the formula;


Computing probability of events

Computing Probability of Events

  • What is the probability in a family with three children that two of the children are girls?

  • Using example 2 that there are eight outcomes in the sample set. G={ggb,gbg,bgg}


Probability theory

  • Therefore, n(G) =3 and n(S) = 8.


Probability theory

  • What is the probability that a total of four shows when we roll two fair dice?

  • n(S) = what? What is the sample space?

  • The sample space for rolling two dice has 36 ordered pairs of numbers.


Probability theory

  • Rolling a four, F = {(1,3),(2,2)(3,1)}

  • Therefore,


Basic properties of probability

Basic Properties of Probability

  • Assume that S is a sample space for some experiment and E is an event in S.

    • 1)

    • 2)

    • 3)


Probability formula for computing odds

Probability formula for computing odds

  • If E’ is the complement of the event E, then the odds against E are


Example problem

Example problem

  • Suppose that the probability of the Saints winning the Super Bowl is 0.15. What are the odds against the Saints winning the Super Bowl.


Probability theory

  • 1-0.15 = 0.85

  • This answer is the complement, P(E’)

  • 0.15 is the probability of E, P(E)

  • The odds against the Saints are


Probability theory

  • Therefore, we would say that the odds against the Saints winning the Super Bowl are 17 to 3.


  • Login