PROBABILITY THEORY

Chapter 14 sec 1

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.'"

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

Aristotle The probable is what usually happens.

Bertrand, Joseph Calcul des probabilités How dare we speak of the laws of chance? Is not chance the antithesis of all law?

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

Caesar, Julius Iacta alea est. (The die is cast.)

Doyle, Sir Arthur Conan The Sign of Four When you have eliminated the impossible, what ever remains, however improbable, must be the truth.

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

Def. Is any observation of a random phenomenon.

Outcome

Def. The different possible results of the experiment

Sample Space

Def. The set of all possible outcomes for an experiment.

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

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

Def. In probability theory, an event is 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

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.

0.5

1.0

0.0

Certain

Likely to occur50-50 Chance of occurring

Impossible

Not likely to occur

Probability of an event (E)

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

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

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

Formula

𝑃ሺ𝐸ሻ= 𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑

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

Support Oppose

In town 1,512 2,268

Surrounding

3,528 1,764

Problem

If a newspaper reporter randomly selects a person to interview from these people, What is the probability that the person supports the racetrack?

9

5

1764226835281512

35281512)(

sP

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

We normally say, 5 to 4

4

5

17642268

35281512

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;

)(

)()(

Sn

EnEP

Computing Probability of EventsWhat 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}

Therefore, n(G) =3 and n(S) = 8.

8

3

)(

)()(

Sn

GnGP

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.

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

Therefore,

12

1

36

3

)(

)()(

Sn

FnFP

Basic Properties of ProbabilityAssume that S is a sample space for some experiment and E is an event in S.

1)2)3)

1)(0 EP0)( P1)( SP

Probability formula for computing odds

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

)(

)'(

EP

EP

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.

1-0.15 = 0.85This answer is the complement, P(E’)

0.15 is the probability of E, P(E)

The odds against the Saints are 3

17

15

85

10015.0

10085.0

15.0

85.0

)(

)'(

x

x

EP

EP

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