Lecture 9 Probability

8/10/2019 Lecture 9 Probability

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Chance and probability

We live in an uncertain world making hundreds

of guesses, calculated risks and somegambles

Important in business

The concept of probability or chance is onethat arises quite naturally we talk of somethings being more common than others, ofrare events, of inevitable happenings, etc.

Very often we are happy to conjure up such

probabilities ourselves thus we are happy toassert that a fair coin has a 50:50 chance oflanding heads, or a fair six-sided die has a onein six chance of giving a 6 when thrown, etc.This is an example of assigning probabilitiesby arguments of symmetry .


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Ground rules - The Axioms ofProbability all probabilities must lie between 0 and

1 (or 0% and 100%); an event that is certain to happen has

probability 1 (or 100%);

in a situation where two events cannotboth occur together, then theprobability that either of them occurs isthe sum of the two individualprobabilities. The two events are called

mutually exclusive

Immediate consequence:

Probability of not occurring

=

1 Probability of occurring


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Allocation of probabilities

Probabilities are usually allocatedto events using arguments of:

symmetry , where all possible outcomes arelisted in such a way that we are happy that allitems in the list are equal ly l ike ly thenthe probability that a particular event occurs is:

p =Number of outcomes in which the event occursTotal number of possible outcomes

geometrical arguments

historical frequency (as in assessment of riskor weather forecasting)

subjective assessments


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FirstCoin

SecondCoin

Probability

1 Head Head

2 Head Tail

3 Tail Head

4 Tail Tail

1

There are four possible equally l ikelyoutcomes when tossing two fair coins.

Three balls are chosen at random fromthis container:

The probability of:

Firstly pulling a red ball out of the bag is4 / 9 = 0.444 = 44.4%

Secondly pulling a yellow ball out of the bag is 2 / 8 =0.25 = 25%

Thirdly pulling a red ball out of the bag is3 / 7 = 0.4286 = 42.9%

1.

.


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366udents)

Marketing B & M Economics A + F

emale 28 62 34 53ale 42 79 29 39

he probability of a student:

Doing a Business and Management degree:P = 62 + 79 = 141 = 0.3852 / 38.5%

366 366

Being Female:= 28 + 62 + 34 + 53 = 177 = 0.4836 / 48.4%366 366

Being Male and doing a Marketing or Economicsegree:

P = 42 + 29 = 71 = 0.194 / 19.4%366 366

.


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Venn diagrams

Two or more probabilities can be added

together only when the probabilities beingadded refer to events that cannot happensimultaneously, mutually exclusive events .

Ideas about such situations may be

clarified by representing eventsdiagrammatically if two events cannotoccur together, then they are representedby non-overlapping areas; if it is possiblefor them to occur together, theirrepresentations should overlap.


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Addition rule forprobabilities

Probability of EITHER of two events=

Probability of the first+

Probability of the second-

Probability of both


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ADDITION RULE

)()()()( B A P B P A P B A P

This is often written

P ( A or B )

= P (A ) + P ( B ) - P ( A and B )

OR


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xample

A card is drawn from an ordinary

pack of playing cards. Find the probabilityhat a card is) a club b) a diamond c) a King) a club or a diamond e) a club or a King

a) 13/52 =

b) 13/52 =

c) 4/52= 1/13

d) Mutually exclusive events so a) + b)13/52 + 13/52 = 26/52 =

e) Not mutually exclusive13/52 +4/52 - 1/52 = 16/52 = 4/13

Using P ( A or B ) = P (A ) + P ( B ) - P ( A and B)or just count the number of relevant cards

13 + 3 = 16 then 16/52 = 4/13


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English Irish Scottish Welsh

18 20 28 6 4 2020 andover

24 4 2 35

surveyonducted byhe Nationalibrary found

he followingesults aboutheir members:

What is the probability that a randomly chosenmember is aged 20 and over and Welsh?

35 / 123 = 0.2846

f the randomly selected member is 18-20 , what ishe probability that they are English?

Number in interval 18-20 = 28+6+4+20 = 58

P (English | 18-20) = 28/58 = 0.4828

(this is called a conditional probability)


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Odds

Odds against=

Probability of not occurringProbability of occurring

xample Two heads in two tosses Probability = = 25%

Odds are 3 to 1 against


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Example : Roulette A standard US roulette wheel is divided into 38

arcs of equal length bearing the numbers00,0,1,2,,35,36 as shown. The number ofthe arc on which the ball stops is the outcomeof one play of the game. The arcs are colouredas follows:

What are the probabilities of obtaining(i) a red score(ii) an even score(iii) an odd black score?

What are the odds against a green score?

1 2 3 45

67

89

10

1112

13

1415

1617181920212223

2425

2627

28

2930

3132

3334

35 36 0 00


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Example: Consumer complaints 1

A manufacturer of electromechanical kitchen utensilsinvestigated 100 complaints received from customers

One complaint was chosen at random. What are theprobabilities that:

(1) It was based on an electrical fault(2) It was for a complaint during the guarantee period(3) It was within the guarantee period given that it was

electrical

Duringguaranteeperiod

Outsideguaranteeperiod

Electrical 18 12

Mechanical13 22

Appearance 32 3


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Sequences of events

When there are sequential events the outcomesof earlier events are likely to affect the laterresults.

The Multiplication Rule is used to find

robability of event A followed by event B

=robability of A x Probability of B (adjustedfor A having occurred if necessary)


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The Multiplication Rule

P( A and B ) = P(A) x P( B|A )

orP ( A B ) = P ( A ) x P ( B|A )

For Independent events

P ( A B ) = P ( A ) x P ( B )


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1. The probability of getting twoconsecutive 3s when throwing a die:

1/6 x 1/6 = 1/36

2. The chance of two consecutive headswhen tossing a fair coin is

=

3. The chance of no sixes in two throwsof a die is

(these events are called independent )

4. The probability of getting 3 consecutiveJacks from a pack of playing cards:

4/52 x 3/51 x 2/50 = 1/5525(these events are not independent )

5 5 250.6944

6 6 36

x


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Consumer complaints - answers

1) both were based on an electrical fault

2) both were for complaints during the guarantee peri

3) both concerned the same type of faultP (EE) + P(MM) + P(AA)=

30 290.0879

100 99

x

63 620.3945

100 99x

30 29 35 34 35 34100 99 100 99 100 99

650.3283198

x + x + x


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xamples

220 to 1220/2211/221TwoAcesCards13,983,815

to 1

13,983,815/13,983,8161/13,983,

816JackpotLottery

7,887,3626,096,454NoLottery

11 to 2511/3625/36Nosixes

2 dice

3 to 13/41/4Twoheads

2 coins

Evens1/21/2Even

score

Die

1 to 51/65/6Not a 6Die

5 to 15/61/66 Die

1:11/21/2HeadCoin

ODDSChanceof NOT

ChanceEventituation

Documents

Lecture 9 Probability