probability explained with animation

Probability

Chapter-3

What should you know to understand the Probability ?

• Addition ,• Subtraction ,• Multiplication ,• Division, and• Values between 0 and 1

What is the chance that the sales will decrease if, the price is

increased ?

What is the chance that Indians will not live after the age of 65 years ?

What is the likelihood that the driving will be safe on Indian

roads ?

Probability

• Likelihood• Chance• Possibility• Odds

• What are the chances that sales decrease if we increase the price ?

• What is the likelihood the new method will result in high productivity ?

• What are the odds in favour of a new investment being profitable ?

• What is the likelihood the driving will be safe on Indian roads ?

• What are the chances that Indian will not live after 65 years of age ?

• What are the chances that TV serials telecasted between 8 & 9 P.m will be seen by the family ?

Probability

• Probability is the chance that an event will occur

• What are the chances that sales decrease if we increase the price ?

• What is the likelihood the new method will result in

high productivity ?• What are the odds in favour of a new investment

being profitable ?

• What is the likelihood the driving will be safe on Indian roads ?

• What are the chances that Indians will not live after 65 years of age ?

• What are the chances that TV serials telecasted between 8 & 9 P.m will be seen by the family ?

Probability

• Probability is the Likelihood that an event will occur

• Probability values are assigned on a scale from 0 to 1

Or • Measured in percentage

Probability

0 10.5

Probability

50:50 Chance

Uncertain Certain

Probability-Quiz

• Match the following with the above Chances1. Sun will arise in the east tomorrow2. It will rain today3. Tomorrow We can travel by Metro Train in Bangalore4. Every Monday morning there will be heavy traffic on the

Bangalore Road 5. Everybody will die at the age of 85

LikelyUnlikely

0 1

50:50 Chance

Uncertain Certain

Probability- Terminology

• Experiment:An activity that takes place

• Outcomes:One of the possible results of an experiment. The experiment will result in exactly one outcome.

• Events:Specifically defined outcome that is of particular interest to us

• Randomly Selecting a Student based on the sex

• Possible outcomes1. Boy2. Girl

Randomly Selecting a Student who is a

boyExperiments OutcomesElection Win , LossExamination Pass , FailSelling Purchase , no PurchasePlaying cricket Win, Loss, TieToss a coin Head , Tail

Probability

Single Event

Single Event Complementay Event

Joining Multiple Events

Mutually Exclusiv

e Even

ts

Addition Rule

Not Mutually Exclusiv

e Even

ts

Addition Rule

Independent Even

t

Multiplication Rule

Conditional

Event

Multiplication Rule

Measuring Probability

Experiment: Roll a die which has six sides with each side showing one number from 1 to 6 and with each side showing a different number.Event:If we define an event as the number 5 showing, what is the probability of this event happening ?

Step-1: Find the Total number of possible outcome of the experiment

we may get either the number 1 or 2 or 3 or 4 or 5 or number 6 on rolling the die. Therefore total number of possible outcomes are 6.

Step-2: Find the number of ways the event could occur

On rolling the die the number 5 will occur in only one way

Step-3: Substitute the values in the formula

P( Number 5 showing ) = 1/6 P( Number 5 showing ) = 0.17

P(Event) = Number of ways the event could occur

Total number of outcomes

If we roll the die once 1) what is the probability of 5 showing ? or 2) what is the chance that the number 5 will be shown ? or 3) How much percentage of an experiment will result in number 5 ?

Interpretation: Since the probability calculated now is 0.17 , theoretically we can say that , 17% of the times, the result of an experiment (of Rolling a die ) will be number 5 and the balance 83% of the times the experiment will result in other numbers.

If we roll the die once 1) what is the probability of 5 showing ? or 2) what is the chance that the number 5 will be shown ? or 3) How much percentage of an experiment will result in number 5 ?

17%

83%

Rolling a Die

Number 5other numbers

Measuring Probability- QuizExperiment:

Roll a die which has six sides with each side showing one number from 1 to 6 and with each side showing a different number.

Event:1. what is the probability of number 1 showing?2. what is the probability of number 2 showing?3. what is the probability of number 3 showing?4. what is the probability of number 4 showing?5. what is the probability of number 6 showing?

• P(Number 1) = 1/6 =0.167

• P(Number 2) = 1/6 =0.167

• P(Number 3) = 1/6 =0.167

• P(Number 4) = 1/6 =0.167

• P(Number 6) = 1/6 =0.167



16.7

16.7

16.716.7

16.7

16.7

Rolling a Die

Number-1Number-2Number-3Number-4Number-5Number-6

All the outcomes are having equal probability or chance. Hence this type of outcomes are called Equally Likely

Probability of all the outcomes of an experiment =1. P(All) = 0.167+0.167+0.167+0.167+0.167+0.167Sample space for this rolling die experiment is : S= [1,2,3,4,5,6]

Set of all the possible outcomes of an experiment is called Sample Space

Complement of an Set of all the possible outcomes of an experiment is called Sample Space

Quiz

• http://www.bbc.co.uk/skillswise/numbers/handlingdata/probability/flash1.shtml

http://www.bbc.co.uk/skillswise/numbers/handlingdata/probability/flash1.shtml

http://www.bbc.co.uk/skillswise/numbers/handlingdata/probability/flash1.shtml



Event:1. what is the probability of showing odd number ?2. what is the probability of showing even number ?

Odd = 1,3,5Even = 2,4,6

• P(Odd Number) = 3/6 =0.5• P(Even Number)= 3/6 =0.5



50%50%

Rolling a Die

Odd NumberEven Number



Event:1. what is the probability of showing numbers greater than 3 ?2. what is the probability of showing numbers less than 5 ?3. What is the probability of number showing more than 3 and less than 6 ?

• P(Greater than 3) = 3/6 =0.5

• P(Less than 5) = 4/6 =0.33



Complement of an Event

Experiment: Roll a die which has six sides with each side showing one number from 1 to 6 and with each side showing a different number.

Event:1. what is the probability of showing numbers greater than 3 ?2. what is the probability of showing numbers less than 5 ?3. What is the probability of number showing more than 3 and less than 6 ?

Event-1. probability of showing numbers greater than 3

Complement of this event is = probability of not showing numbers greater than 3

P(Number showing >3 ) = 3/6 = 0.5P(Not showing number>3) = 1-0.5

=0.5

Event-2. probability of showing numbers less than 3

Complement of this event is = probability of not showing numbers less than 3

P(Number showing <3 ) = 2/6 = 0.33P(Not showing number<3) = 1-0.33

= 0.67

The probability for the opposite of an event E is called Complement of that event . Complement is denoted as E’.

P(E’)= 1-P(E)

A A’

Joining Two Or more Events

• Addition Rule

Mutually Exclusive EventsMutually exclusive

If event A happens B can not happen Or Vice-versa

A B C D

Mutually Exclusive Events Not Mutually Exclusive Events

Mutually exclusive If two events do not have common outcome the events are mutually Exclusive

Mutually Exclusive EventsMutually exclusive

If event A happens B can not happen .They will not happen at the same time They will not have no common outcome

A B

Mutually Exclusive Events Not Mutually Exclusive Events

Not Mutually Exclusive If two events have common outcome the events are Not mutually Exclusive

C D

A. Sun arises in the morning . B. Sun does not arise in the morning

Hat is the

What is the probability of selecting a King ?

What is the probability of selecting a Queen ?What is the probability of selecting a Queen or King ?

What is the probability of selecting a Queen or Black ?

Joint Probability - Quiz

• The Outcome of Event E= {3,4,6}Event F= {2,5,7}Event G= {1,2,5}Event H= {8,9,6}Find the following are mutuallyexclusive or not:1. E & F 2. F&G 3. G&H 4. E&G 5. E&H 6. F&H7. Events that an individual is: A: Married, B: Single, C: Divorced

Find the result of joining the following events:

1. E&F 2. F&G 3. G&H 4. E&G 5. E&H 6. F&H 7. F or E 8. F or G 9. G or H10. E or G11. E or H12. F or H13. The symbol ‘U’ means 14. The symbol ‘ ’ means

Mutually Exclusive

Not Mutually Exclusive


Mutually ExclusiveMutually Exclusive

Mutually Exclusive

= Nil

= Nil

= Nil

= Nil

= {2,5}

= {6}

= {2,5,7,3,4,6}= {2,5,7,1}

= {1,2,5,8,9,6}= {2,5,7,3,4,6}= {3,4,6,8,9}

= {2,5,7,8,9,6}

What will be the result if either of the events happen at a time ?

The Addition Rule-1

Mutually Exclusive

1. P(A U B) = P(A) + P(B)P(A or B) = P(A) + P(B)

2. P(A or B or C) = P(A)+P(B)+P(C )


1. P(A U B) = P(A) + P(B) – P(A B)P(A or B ) = P(A) + P(B) – P(A and B)

2. P(A or B or C ) = P(A) + P(B) +P(C) – P(A and B) - P(A and C) – P(C and B) –P(A and B and C)

All Red cards without Number 2 red Cards and All black number2 cards

Red Card

Number 2 Red Card

Black Card

All read and All Black Cards

Questions: Either A or B

What will be the result if either of the event to happen at a time ?

Remove no.2 red cards

Addition Rule-1 -Quiz

1. What is the probability that it is either a king or a Queen ?

2. What is the probability that it is either a king or a Ace ?

3. What is the probability that it is either a Red or number 1 ?

4. What is the probability that it is either a Ace or a number2 ?

5. What is the probability that it is either no.1 or no.2 ?

6. What is the probability that it is either a king or Red ?

Which of the following is Mutually Exclusive event and match the formula on the opposite :

One Card is selected from 52 cards:

3. P(A U B) = P(A)+P(B)

4. P(A U B) = P(A) + P(B) – P(A B)

2. P(A or B) = P(A)+P(B)

1. P(A or B ) = P(A) + P(B) – P(A and B)

The Addition Rule-2

Mutually Exclusive

1. P(A B) = 0P(A and B) = 0

2. P(A and B and C) = 0


1. P(A B) = P(A) + P(B) – P(A U B)P(A and B ) = P(A) + P(B) – P(A or B)

2. P(A and B and C ) = P(A) + P(B) +P(C) – P(A or B) - P(A or C) – P(C or B) –P(A or B or C)

Questions: Both A and B

What will be the result if both events to happen at a time ?

Red Card

Black Card

Both read + Black Card will not be available

Red and Two

Red Card

Number 2

Addition Rule-2 -Quiz

1. What is the probability that it is both a king and Queen ?

2. What is the probability that it is both a king and Black ?

3. What is the probability that it is both a Red or number 1 ?

4. What is the probability that it is both Ace or a number2 ?

5. What is the probability that it is both number no.1 and no.2 ?

6. What is the probability that it is both a king and Red ?

Which of the following is Mutually Exclusive event and match the formula on the opposite :

One Card is selected from 52 cards:

3. P(A B) = 0

4. P(A B) = P(A) + P(B) – P(A U B)

2. P(A and B ) = 0

1. P(A and B ) = P(A) + P(B) – P(A or B)

Probability of Joint Events

• What is the probability of number showing more than 3 and less than 6 ?

• Step-1: Find the outcomes of event-1 &2• Let Event-1 = Probability of showing more than 3 • Let Event-2 = Probability of showing less than 6• We name the event-1 as E and event-2 as F• Outcomes of E = { 4,5,6}• Outcomes of F = { 1,2,3,4,5}• Step-2: Find the common outcomes of event-1 &2

• Common outcomes to both event 1&2 = 4,5• Step-3: Find whether the events 1&2 are mutually

exclusive or not mutually exclusive:• Our events are not mutually exclusive. Hence we have to apply

the suitable formula

E

4,5,6

F

1,2,3, 4,5

E

6

F

4,5 1,2,3

Two events of an experiment can be joined using ‘And’ , ‘Or’



• Step- 4: Decide the suitable formula:

• 1.If events are mutually exclusive the formula for ‘and’ is:• P(A B) = 0• 2.If events are not mutually exclusive the formula for ‘and’ is:

• P(A B) = P(A) + P(B) – P(A U B)• Our events are not mutually exclusive. Hence we have

to apply the second formulaP(A B)

Probability of A and BP(A AND B)Probability of A intersection B

P(A U B)Probability of A OR BP(A OR B)Probability of A Union B



• Step- 5: Find the P(A):• P(A) = P( Number showing more than 3) = 3/6 = 0.5• Step- 6: Find the P(B):• P(B) = P( Number showing less than 6) = 5/6 = 0.83• Step- 7: Find the P(A OR B):• P(AUB) = P( Number showing all the outcomes of more than 3 & less than 6)

= 6/6 = 1• Step- 8: Apply the values in to the formula:• P(A B) = P(A) + P(B) – p(AUB)

• = 0.5 + 0.83 – 1• = 0.33

E

4,5,6

F

1,2,3, 4,5

E OR F

1,2,3, 4,5,6

P ( A and B) = 0.33 .P ( number showing more than 3 and less than 6 ) = 0.33P ( Outcomes common to events of showing number more than 3 and less than 6) = 0.33P ( showing 4,5 is ) = 0.33


• What is the probability of number showing more than 3 or less than 6 ?

• Step-1: Find the outcomes of event-1 &2• Let Event-1 = Probability of showing more than 3 • Let Event-2 = Probability of showing less than 6• We name the event-1 as E and event-2 as F• Outcomes of E = { 4,5,6}• Outcomes of F = { 1,2,3,4,5}• Step-2: Find the combined outcomes of event-1 &2

• Combined outcomes to both event 1&2 = 1,2,34,5,6• Step-3: Find whether the events 1&2 are mutually

exclusive or not mutually exclusive:• Our events are not mutually exclusive. Hence we have to apply

the second formula

Probability of Joint Events – using ‘OR’

• What is the probability of number showing more than 3 Or less than 6 ?

• Step- 4: Decide the suitable formula:

• 1.If events are mutually exclusive the formula for ‘Or’ is:• P(A U B) = P(A) + P(B)• 2.If events are not mutually exclusive the formula for ‘ Or’ is:

• P(A U B) = P(A) + P(B) – P(A B)


• What is the probability of number showing more than 3 Or less than 6 ?

• Step- 5: Find the P(A):• P(A) = P( Number showing more than 3) = 3/6 = 0.5• Step- 6: Find the P(B):• P(B) = P( Number showing less than 6) = 5/6 = 0.83• Step- 7: Find the P(A AND B):• P(A B) = P( Number showing common outcomes to the events> 3 & < 6)

= 2/6 = 0.33• Step- 8: Apply the values in to the formula:• P(A U B) = P(A) + P(B) – p(A B)

• = 0.5 + 0.83 – 0.33• = 1

4,5,6

P ( A or B) = 1P ( number showing more than 3 or less than 6 ) = 1P ( All outcomes of events of showing number more than 3 and less than 6) = 1P ( showing 1,2,3,4,5,6 is ) = 1

Addition using Neither ….Nor QuizThe Outcome of Event E= {3,4,6} Event F= {2,5,7}Event G= {1,2,5}Event H= {8,9,6}

1. E&F 2. F&G 3. G&H 4. E&G 5. E&H 6. F&H

= Nil

= Nil

= Nil

= Nil

= {2,5}

= {6}

= {3,4,6,2,5,7}

= {1,2,5,8,9,6}

= {3,4,6,1,2,5 }

= {2,5,7,8,9,6 }

= {1,7}

= {3,4,8,9}

1. Neither E nor F

2. Neither F nor G

3. Neither G nor H

4. Neither E nor G

5. Neither E nor H

6. Neither F nor H

What will be the result if either of the events to happen at a time ?

What will be the result if Neither of the events to happen at a time ?

The Addition Rule-2

Mutually Exclusive

1. P(A U B) = 0P(neither A nor B) = 0


1. P(A U B) = 1- P(AUB) =1- { P(A) + P(B) – P(A

B)}P(neither A nor B ) =1- { P(A) + P(B) – P(A and B)}

Questions: Neither A Nor B

Multiplication Rule

Independent & Conditional EventsIndependent Event

• Example:• The probability of a person

buying a MBA Scanner is not affected by sex.

• Hence the event ‘Buying a MBA Scanner’ is not affected by the other event ‘ male/female buyer’

• Hence the event ‘Buying a MBA Scanner’ is independent of the event ‘ buyer being male / female’

Conditional Event

• Example:• The probability of a person

buying a MBA Scanner is affected by the specialisation.

• Hence the event ‘Buying a MBA Scanner’ is affected by the other event ‘marketing /Finance branch of the buyer’

• Hence the event ‘Buying a MBA Scanner’ dependents on the event ‘marketing /Finance branch of the buyer’

The probability of an event occurring will be get affected by other events or extra information.

The probability of an event occurring will not be get affected by other events.

Conditional Probability

• In a game suppose your opponent has thrown a dice and you have not seen the result . Throwing 6 is the winning criteria .

We define :Event A: Throw is a number 61. P(Throw is a number 6)= 1/6 = 16%2. Opponent has 16% chance to winNow the opponent says he had thrown an even

number. Hence we define :

Event B: Throw is an even number 3. Possible outcomes are 2,4,64. 6 is one among the outcomes – Event A5. The conditional probability of throwing 6 the

event-A after we know that the throw is even is = 1/3 = 33%

6. Now Opponent has 33% chance to win

The probability that the event ‘A’ occurs when we know that ‘B’ has occurred is conditional probability

Conditional Events

• If we have two events A & B the probability P (A|B) is the Conditional probability of A given B.• That is the probability that the event

A occurs when we know that B has occurred. • |- this vertical line means “given”• The event on the right of the vertical line is the additional

information.

Independent & Conditional Events - Quiz

• 1. An individual has high IQ. An individual is selected for university

post

• 2. A patient takes long time to recover from an operation. The patient

is elderly

• 3. A student plays chess. A student is good at Maths

• 4. A student plays Table - Tennis. A student is good at Maths

• 5. Today it rains. Today is Tuesday

• 6. A company appoints its CMD. Successful candidate is women

Find Independent or Conditional Events

Multiplication Rule• Independent EventsP(A and B) = P(A) * P(B)

• Conditional Events P(A and B) = P(A|B)* P(B)

Two mutually exclusive events cannot be independent

Calculating Independent & Conditional Probabilities

• Independent Probability• If two events A & B are

independent then the probability that they both occur is:

• Conditional Probability• If two events are dependent

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

P(A and B)= P(A)* P(B)

P(A, B and C )= P(A)* P(B) * P(C) P(A and B)= P(A)* P(A|B)

probability explained with animation

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