ProbabilityProbability

Math 374Math 374

Game PlanGame Plan

GeneralGeneral ModelsModelsa)a) Tree DiagramTree Diagramb)b) Matrix Two Dimensional ModelMatrix Two Dimensional Modelc)c) Balanced Balanced d)d) Unbalanced Unbalanced Odds – for – odds againstOdds – for – odds against

What is ProbabilityWhat is Probability

It is a number we assign to It is a number we assign to show the likelihood of an show the likelihood of an event occurringevent occurring

We set the following limitsWe set the following limits What is the probability that if I What is the probability that if I

drop the piece of chalk it will drop the piece of chalk it will fall to the floor?fall to the floor?

P (fall) = 1 a certainlyP (fall) = 1 a certainly

ProbabilityProbability

What is the probability that What is the probability that the chalk will float up to the the chalk will float up to the ceiling?ceiling?

P (float) = 0 an impossibilityP (float) = 0 an impossibility

Probability ScaleProbability Scale

We have created a scaleWe have created a scale

0

Absolute Impossibility

1

Absolute

Certainty

Various Types of Various Types of ProbabilityProbability

Subjective – gets you in troubleSubjective – gets you in trouble Probability – (Canadiens will will Probability – (Canadiens will will

Stanley Cup)Stanley Cup)

Experimental – you need to do an experimentExperimental – you need to do an experiment Probability (cars on an assembly line have a bad Probability (cars on an assembly line have a bad

headlight). You would probably test 20 cars. If 1 headlight). You would probably test 20 cars. If 1 was faulty you would say 1/20 are badwas faulty you would say 1/20 are bad

0 1

0.1 A leafs fan

0.8 (A fan)

Various Types of Various Types of ProbabilityProbability

Theoretical – the one we will Theoretical – the one we will useuse

Fundamental DefinitionFundamental Definition P = P = SS

RR

where s # of successeswhere s # of successes

R # of possibilities R # of possibilities

ExamplesExamples

Consider flip a coin, what is the Consider flip a coin, what is the probability of getting a tail probability of getting a tail

S = (T) = 1S = (T) = 1

R = (H, T) = 2 R = (H, T) = 2

P = ½P = ½

ExamplesExamples

Roll a die, get a 5Roll a die, get a 5 S = (5) = 1 S = (5) = 1 R = (1,2,3,4,5,6) = 6R = (1,2,3,4,5,6) = 6 P = 1/6 P = 1/6 Roll a die, get more than 2Roll a die, get more than 2 S = (3,4,5,6)S = (3,4,5,6) R = 6 R = 6 P = 4/6 P = 4/6 (you do not need to reduce in this (you do not need to reduce in this

chapter!)chapter!)

ModelsModels The key to understanding probability The key to understanding probability

is to have a model that shows you is to have a model that shows you the possibilities the possibilities

This can get daunting, there are 311 This can get daunting, there are 311 875 200 possible poker hands from a 875 200 possible poker hands from a standard deck.standard deck.

The easiest model we will use is a The easiest model we will use is a treetree

Tree ModelTree Model - Flipping two coins - Flipping two coins

H

TT

H

T

H

Starting Point

We need to Determine RWe need to Determine R In a balanced model just count the In a balanced model just count the

number of end branches i.e. 4 to number of end branches i.e. 4 to determine denominatordetermine denominator

OR 2) R = # of possibilities of first. # of OR 2) R = # of possibilities of first. # of possibilities of second. # of possibilities possibilities of second. # of possibilities of third.of third.

2 x 2 = 4 2 x 2 = 4 Using the modelUsing the model P (getting two tails)P (getting two tails) S S How many branches from start to How many branches from start to

the end satisfy?the end satisfy? Let’s look at the various types of modelsLet’s look at the various types of models

Tree ModelTree Model

H

T

T

H

T

H

Starting

Point

S = ?

S = 1P = ¼ P

= ?

Notice # of branches will be the denominatorLook at the # of successes for numerator

Matrix Two Dimensional Matrix Two Dimensional ModelModel

Rolling Two Die or Rolling Two Die or DiceDice

Not a treeNot a tree Called a matrix – two Called a matrix – two

dimensional dimensional Eg P (getting a total 5)Eg P (getting a total 5) S = 4S = 4 P = 4/36P = 4/36 Roll over 3 Roll over 3 Do not include 3Do not include 3 P = 33/36P = 33/36

11 22 33 44 55 66

11 22 33 44 55 66 77

22 33 44 55 66 77 88

33 44 55 66 77 88 99

44 55 66 77 88 99 1100

55 66 77 88 99 1100

1111

66 77 88 99 1100

1111

1122

Die #1

Die #2

Balanced ModelBalanced Model Consider a bag with 2 blue Consider a bag with 2 blue

marbles and 3 red marbles. You marbles and 3 red marbles. You are going to pick two and are going to pick two and replace them.replace them.

Replace = put them backReplace = put them back What is the prob of getting a What is the prob of getting a

blue & red?blue & red?

Balanced ModelBalanced Model

B

B

R

R

R

BB

RBB

R

R

R

RR

Starting Point

BB

BB

BB

R

R

R

RRR

# of successes? 12

P = 12/25

# of Possibilities? R = 25

P (blue & Red)?

R

RR

Put check marks!

Unbalanced ModelUnbalanced Model It is not always possible to write It is not always possible to write

out every single branch. out every single branch. Consider the same question;Consider the same question;

What is the P of getting a blue What is the P of getting a blue and a red?and a red?

This time we create an This time we create an unbalanced modelunbalanced model

Starting Point

2

3

B

B

R

R

R

B

2

2

3

3

S?(2x3)+(3x2)R?

P=12/25

R=5x5

To find den. ADD branches and MULT each one. (It differs if you have 3 options).

Unbalanced ModelUnbalanced Model Create a model given a bag with 20 blue, Create a model given a bag with 20 blue,

15 green and 15 red marbles. You are 15 green and 15 red marbles. You are picking three marbles and replacing them. picking three marbles and replacing them.

What is the probability of getting three What is the probability of getting three green?green?

Draw the model!Draw the model! S = ?S = ? 15 x 15 x 1515 x 15 x 15 R = ?R = ? 50 x 50 x 5050 x 50 x 50 P = 3375 / 125000P = 3375 / 125000

Unbalanced ModelUnbalanced Model What is the probability of getting a What is the probability of getting a

blue, a green and a red?blue, a green and a red? Since they do not mention it, we must Since they do not mention it, we must

assume order does not matter.assume order does not matter. We need to look at BGR, BRG, GRB, We need to look at BGR, BRG, GRB,

GBR, RBG and RGB. GBR, RBG and RGB. S = (20x15x15) + ?S = (20x15x15) + ?

+ (20x15x15) + (15x15x20) + + (20x15x15) + (15x15x20) + (15x20x15) + (15x20x15) + (15x20x15) + (15x20x15) + (15x15x20) = 27000(15x15x20) = 27000

P = 27000 / 125000P = 27000 / 125000

Without ReplacementWithout Replacement

Without replacement = not Without replacement = not putting them back (you have less putting them back (you have less possibilities afterwards)possibilities afterwards)

Given a bag with 5 red, 10 blue Given a bag with 5 red, 10 blue and 15 green and you will pick and 15 green and you will pick three marbles and do not replace three marbles and do not replace them.them.

Create a modelCreate a model

Without ReplacementWithout Replacement What is the probability of getting a What is the probability of getting a

B-R-G in any order? (5 red, 10 blue B-R-G in any order? (5 red, 10 blue and 15 green) and 15 green)

So we are looking at RBG, RGB, So we are looking at RBG, RGB, BRG BGR GRB GBRBRG BGR GRB GBR

S = (5x10x15) + (5x15x10) + S = (5x10x15) + (5x15x10) + (10x5x15) + (10x15x5) + (15x5x10) (10x5x15) + (10x15x5) + (15x5x10) + (15x10x5) = 4500+ (15x10x5) = 4500

R = ?R = ? P = 4500 / 24360P = 4500 / 24360

R = 30 x 29 x 28 = 24360

Without ReplacementWithout Replacement

What is theWhat is the probability of getting 2 B probability of getting 2 B and one G or two G and one B?and one G or two G and one B?

So we are looking at BBG BGB GBB So we are looking at BBG BGB GBB GGB GBG BGGGGB GBG BGG

S = (10x9x15) + (10x15x9) + (15x10x9) S = (10x9x15) + (10x15x9) + (15x10x9) + (15x14x10) + (15x10x14) + + (15x14x10) + (15x10x14) + (10x15x14) = 10350(10x15x14) = 10350

P = 10350 / 24360P = 10350 / 24360 Do Stencil #5,6,7 Do Stencil #5,6,7

Odds For – Odds AgainstOdds For – Odds Against Another way of showing a situation in Another way of showing a situation in

probability is by oddsprobability is by odds Note: These are not bookie odds – that Note: These are not bookie odds – that

is subjective probability!is subjective probability! We have so far P = We have so far P = SS

RR We will now define F as the number of We will now define F as the number of

failures. Thus S + F = Rfailures. Thus S + F = R # of Successes + # of Failures = # of # of Successes + # of Failures = # of

PossibilitiesPossibilities

Odds ForOdds For Odds for are stated S : FOdds for are stated S : F Eg The odds for flipping a coin and Eg The odds for flipping a coin and

getting a head is 1:1 getting a head is 1:1 Eg The odds for flipping two coins Eg The odds for flipping two coins

and getting two heads 1:3 and getting two heads 1:3

Odds AgainstOdds Against Odds against are stated F : SOdds against are stated F : S Eg The odds against flipping two Eg The odds against flipping two

coins and getting two heads coins and getting two heads 3:1 3:1 If the odds for an event are 8:3, If the odds for an event are 8:3,

what is the probability?what is the probability? S = 8, F = 3 Thus R = 8 + 3 = 11S = 8, F = 3 Thus R = 8 + 3 = 11 P = 8 / 11P = 8 / 11

Last Question Last Question If the odds against are 9:23, what If the odds against are 9:23, what

are the odds for and probability are the odds for and probability 23:923:9 P = 23/32P = 23/32 Do Stencil #8, & #9 Do Stencil #8, & #9