itc lecture #01

8/21/2019 ITC Lecture #01

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Introduction to Probability

It is the science in which either we study a

random experiment or we observe a random

phenomenon.

In probability study, a sample space is needed

which is the set of all possible outcomes of

any random experiment.

It is the connectivity b/w Descriptive and

Inferential Statistics.


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Logical Thinking motivation

Drawing a FISH can help us understand the logicalthinking:

Now, try to re-draw the same fish, but withoutlifting your pen once it touches the paper andwithout striking out any of your drew line.


3/23

Logical Thinking through the Venn

diagram

A Venn diagram is a rectangular area showing

the Sample Space & having some circles inside

(usually overlapped) which are showing the

Events.

S={a,b,c,d,.,n}

A={a,b,c,f,g,h}

B={c,d,e,g,h,i}

C={f,g,h,I,j,k}

a,bc d,e

f

g,h

i

J,kl,m,n

A B

C

S


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Shading the Venn Diagram

For AB, it should be

A B

C

S


For A , it should be


For AB , it should be

The DemorgansLaw


5/23

Probability Topics Tree

Random

Experiment

SampleSpace

Events

Probability

Outcomes Criteria Numeric

Random

Variable

Probability

DistributionExpectation

Mutually Exclusive (Non Overlapping)

Non Mutually Exclusive (Overlapping)

Independent

P(AB)=P(A) P(B)

Dependent

Conditional Probability

Counting Rules


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What is the Distribution?

Gives us a picture of

the variability

and central tendency.

Can also show the

amount of skewness

and Kurtosis.
http://images.google.com/imgres?imgurl=www.surgical-tutor.org.uk/pictures/diagrams/normal_dist.gif&imgrefurl=http://www.surgical-tutor.org.uk/core/neoplasia/statistics.htm&h=575&w=628&sz=8&tbnid=FjJxyA6fz0cJ:&tbnh=121&tbnw=133&prev=/images?q=normal+distribution&hl=en&lr=&ie=UTF-8&oe=UTF-8&sa=G


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Bell-Shaped Symmetrical Distribution

Central Tendency

Dispersion

2

3


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Probability Distributions

For any frequency distribution, we need a

variable while for any probability distribution,

we need a random variable Random Variable is the data which can be

obtained by converting the outcomes of any

sample space into numeric codes after defining

a particular criteria, so;

Random Experiment is necessary for a

probability distribution


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Any Experiment with uncertain results(outcomes) called a random experiment

For example, mixing acid and base will

produce salt and water (Its an experiment)but;

Tossing a Dice or a Coin, or Drawing a cardfrom well shuffled deck will produce a random

result (these are examples of randomexperiments), so in each random experiment,we collect all possibilities (outcomes) andmake a sample space


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Formation of Sample Spaces

Random Experiments Related to a Fair Coin:Random Experiment # 1:Tossing a fair-coin once

S={H,T} 21=2 outcomes

Random Experiment # 2:Tossing a fair coin twice or tossing 2

fair coins, once.

S={HH, HT, TH, TT} 22=4 outcomes

Random Experiment # 3:Tossing a fair coin thrice or tossing 3fair coins, once.

S={HHH, HHT, HTH, THH, THT, TTH, HTT, TTT} 23=8 outcomes

In general, 2nshowing the two sided coin is being tossed n times


11/23

Formation of Dichotomous SS

A truth Table can help us forming the samplespace: For e.g. Sample Space of Rand. Exp. # 3.

The formation rule is simple

Values of Every next column

should be doubled of the

preceding column.

Outcomes can be observed

Horizontally.

S. No. 1st 2nd 3rd

1 H H H

2 T H H

3 H T H

4 T T H

5 H H T

6 T H T

7 H T T

8 T T T


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Random Experiments with Dice

Random Experiment #4:Tossing a fair dice, once

S={1,2,3,4,5,6} 61=6 outcomes

Random Experiment #5:Tossing a fair dice, twice orTossing two fair dice once

S={11, 12, 13, 14, 15,16

21, 22, 23, ,26.....

61, 62, 63, ., 66} 62=36 outcomes
http://www.google.com.pk/imgres?imgurl=http://openclipart.org/people/badaman/badaman_dice.svg&imgrefurl=http://openclipart.org/detail/26713&usg=__ebO8CYxKHWIZveDZ20NqsyuI_NU=&h=367&w=349&sz=11&hl=en&start=12&zoom=1&tbnid=ZRbpI_5SCQJ6ZM:&tbnh=122&tbnw=116&ei=Cil9T8XCGoS8rAfmzKDQDA&prev=/search?q=dice&hl=en&biw=1280&bih=705&gbv=2&tbm=isch&itbs=1


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Random Experiments Contd..

Random Experiment #6:Tossing a fair coin and a fairdice, once

S={H1,H2,H3,H4,H5,H6,T1,T2,.T6} 21 x61=12 outcomes

Random Experiment #7:Tossing 2 fair coins & a fair diceonce.

S={HH1,HH2,HH3,HH4,HH5,HH6

HT1,HT2,HT3,,HT6..

TT1,TT2,.,TT6} 22x61=24 outcomes


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Random Experiments A Deck of Cards

Random Experiment #8: Drawing a card from a

well shuffled Deck of playing cards.

S={ Hearts King+Queen+Jack+Ace+2+3++10 13

Diamonds King+Queen+Jack+Ace+2+3++10 13

Clubs King+Queen+Jack+Ace+2+3++10 13

Spades King+Queen+Jack+Ace+2+3++10} 13

Total= 52


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Formation of Events

What is an Event? Its a logical statement which should be followed, strictly

We always collect the matching outcomes from the samplespace after viewing the Event statement.

For e.g. if we consider the Random Exp. # 2:

Object: Tossing a fair coin twice, S={HH,HT,TH,TT}

Event(s):

A={First toss should be a Head}

A={HH, HT}

B={Exactly one Tail in the outcome}: B={HT,TH}

Thus we formed two Non-Mutually Exclusive Events

HTHH TH

TT

Replicate the same work for

Random Experiment #3

A B

VENN Diagram


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Computing Probability

Probability of an Event P(A)stands for probability of an Event A such that;

P(A) = n(A)/n(S)

Where,

n(A)is the number of outcomes present in Event A.

n(S) are the number of outcomes present in theSample Space.

Probability is a proportion of Event in a Sample Space.

For any Event A; 0 P(A) 1where A S


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Computing Probabilities (Example)

Random Experiment # 2: Tossing a fair coin twice or

tossing two fair coins, once.

Sample Space S={HH,HT,TH,TT},

Event(s)

A={First toss should be a Head}, A={HH, HT}

B={Exactly one Tail in the outcome}: B={HT, TH}

Therefore Probabilities will be,P(A)=2/4=0.5 50% chances

P(B)=2/4=0.5 50% chances


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Interpreting Probability

Probability occurs against every Event and should be interpretedin 3 components;

1) Object of the Random Experiment

2) Value of the Probability

3) Event StatementFor e.g., Interpretation of P(A)=0.5 can be written as;

If we toss a fair coin twice, we have 50% chancesof getting head in the first toss.

Similarly, P(B)=0.5 would be:If we toss a fair coin twice, we have 50% chancesof getting exactly one tail in both tosses.


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Union, Intersection and Compliment

For the same Random Experiment # 2, the followingoperations showing results and relevant interpretationsneeded (where U=OR, =AND, A=not(A):

Since S={HH,HT,TH,TT} A={HH,HT} B={HT,TH}

Therefore,AUB={HH,HT,TH} P(AUB)=3/4=0.75 75%

If we toss a fair coin twice, we have 75% chances of gettinghead in the first toss ORexactly one Tail in both tosses.

AB={HT} P(AB)=1/4=0.25 25%A=S-A={TH,TT} P(A)=2/4=0.50 50%

P(A)=1-P(A)


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Practice Questions

Q1) If we toss a fair coin three times, determine the

following probabilities:

a) P(A)=Probability of getting exactly one Head in all tosses?

b) P(B)=Probability of getting Tail in the first toss?c) P(C)=Probability of getting exactly one head AND one

tail? P(One head One Tail)

d) P(D)=Probability of NOT getting exactly one head in all

tosses? P(A)

e) P(F)=Probability of Either getting exactly one head in all

tosses ORtail in the first toss? P(AUB)


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Practice Questions (Contd..)

Q2) If we toss a fair dice twice, determine the following

Probabilities: (Ref. Random Experiment #4)

a) P(A)=Probability of getting same number on both Dice?

b) P(B)=Probability of getting odd number in both Dice?c) P(C)=Probability of getting sum of both numbers equals

to 5?

d) P(D)=Probability of getting an odd number AND an even

number on two Dice respectively.

e) P(F)=Probability of NOT getting the same number on

both Dice?


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Q3) If we toss a fair COIN and a Fair DICE once, determine

the following Probabilities:(Ref. Random Experiment #6)

a) P(A)=Probability of getting exactly One head in the coin?b) P(B)=Probability of getting an odd number on Dice?

c) P(C)=Probability of getting exactly one Head with an Odd

number on Dice? P(AB)

d) P(D)=Probability of getting a number less than 4 on Dice.

e) P(F)=Probability of NOT getting exactly one Head in the

coin? P(A)=1-P(A)


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Q4) If we toss two fair COINS and a Fair DICE once,determine the following Probabilities: (Ref. RandomExperiment #7)

a) P(A)=Probability of getting exactly One head in the coin?

b) P(B)=Probability of getting an odd number on Dice?

c) P(C)=Probability of getting exactly one Head with an Oddnumber on Dice? P(AB)

d) P(D)=Probability of getting a number less than 4 on Dice.

e) P(F)=Probability of NOT getting exactly one Head in thecoin? P(A)=1-P(A)

itc lecture #01

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