Probability :Probability Chapter-3


What should you know to understand the Probability ? :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 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 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 ? :What is the likelihood that the driving will be safe on Indian roads ?


Probability :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 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 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 :Probability


Probability-Quiz :Probability-Quiz Match the following with the above Chances Sun will arise in the east tomorrow It will rain today Tomorrow We can travel by Metro Train in Bangalore Every Monday morning there will be heavy traffic on the Bangalore Road Everybody will die at the age of 85


Probability- Terminology :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 outcomes Boy Girl Randomly Selecting a Student who is a boy


Measuring Probability :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 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 ?


Measuring Probability- Quiz :Measuring Probability- Quiz 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 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 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+0.167 Sample 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 :Quiz http://www.bbc.co.uk/skillswise/numbers/handlingdata/probability/flash1.shtml


Measuring Probability- Quiz :Measuring Probability- Quiz 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 odd number ? 2. what is the probability of showing even number ? Odd = 1,3,5 Even = 2,4,6 P(Odd Number) = 3/6 =0.5 P(Even Number)= 3/6 =0.5


Measuring Probability- Quiz :Measuring Probability- Quiz 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 ? P(Greater than 3) = 3/6 =0.5 P(Less than 5) = 4/6 =0.33


Complement of an Event :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 = 0.5 P(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 = 0.33 P(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 :Joining Two Or more Events Addition Rule


Mutually Exclusive Events :Mutually Exclusive Events Mutually exclusive If event A happens B can not happen Or Vice-versa Mutually Exclusive Events Not Mutually Exclusive Events Mutually exclusive If two events do not have common outcome the events are mutually Exclusive


Mutually Exclusive Events :Mutually Exclusive Events Mutually exclusive If event A happens B can not happen . They will not happen at the same time They will not have no common outcome Mutually Exclusive Events Not Mutually Exclusive Events Not Mutually Exclusive If two events have common outcome the events are Not mutually Exclusive A. Sun arises in the morning . B. Sun does not arise in the morning


Hat is the :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 :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 mutually exclusive or not: E & F F&G G&H E&G E&H F&H Events that an individual is: A: Married, B: Single, C: Divorced Find the result of joining the following events: E&F F&G G&H E&G E&H F&H F or E F or G G or H E or G E or H F or H The symbol ‘U’ means The symbol ‘ ’ means Mutually Exclusive Not Mutually Exclusive Not Mutually Exclusive Mutually Exclusive Mutually 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 :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 ) Not Mutually Exclusive 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 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 :Addition Rule-1 -Quiz What is the probability that it is either a king or a Queen ? What is the probability that it is either a king or a Ace ? What is the probability that it is either a Red or number 1 ? What is the probability that it is either a Ace or a number2 ? What is the probability that it is either no.1 or no.2 ? 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:


The Addition Rule-2 :The Addition Rule-2 Mutually Exclusive 1. P(A B) = 0 P(A and B) = 0 2. P(A and B and C) = 0 Not Mutually Exclusive 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 and Two


Addition Rule-2 -Quiz :Addition Rule-2 -Quiz What is the probability that it is both a king and Queen ? What is the probability that it is both a king and Black ? What is the probability that it is both a Red or number 1 ? What is the probability that it is both Ace or a number2 ? What is the probability that it is both number no.1 and no.2 ? 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:


Probability of Joint Events :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’


Probability of Joint Events :Probability of Joint Events What is the probability of number showing more than 3 and less than 6 ? 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 formula P(A B) Probability of A and B P(A AND B) Probability of A intersection B P(A U B) Probability of A OR B P(A OR B) Probability of A Union B


Probability of Joint Events :Probability of Joint Events What is the probability of number showing more than 3 and 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 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 ) = 0.33 P ( Outcomes common to events of showing number more than 3 and less than 6) = 0.33 P ( showing 4,5 is ) = 0.33


Probability of Joint Events :Probability of Joint Events 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’ :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)


Probability of Joint Events :Probability of Joint Events 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) = 1 P ( number showing more than 3 or less than 6 ) = 1 P ( All outcomes of events of showing number more than 3 and less than 6) = 1 P ( showing 1,2,3,4,5,6 is ) = 1


Addition using Neither ….Nor Quiz :Addition using Neither ….Nor Quiz The Outcome of Event E= {3,4,6} Event F= {2,5,7} Event G= {1,2,5} Event H= {8,9,6} E&F F&G G&H E&G E&H 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} Neither E nor F Neither F nor G Neither G nor H Neither E nor G Neither E nor H 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 :The Addition Rule-2 Mutually Exclusive P(A U B) = 0 P(neither A nor B) = 0 Not Mutually Exclusive 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 :Multiplication Rule


Independent & Conditional Events :Independent & Conditional Events Independent 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 :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 6 P(Throw is a number 6)= 1/6 = 16% Opponent has 16% chance to win Now the opponent says he had thrown an even number. Hence we define : Event B: Throw is an even number Possible outcomes are 2,4,6 6 is one among the outcomes – Event A The conditional probability of throwing 6 the event-A after we know that the throw is even is = 1/3 = 33% 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 :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 :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 :Multiplication Rule Independent Events P(A and B) = P(A) * P(B) Conditional Events P(A and B) = P(A|B)* P(B)


Slide 42:Two mutually exclusive events cannot be independent


Calculating Independent & Conditional Probabilities :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)