DM Slides - 26 July

Venn DiagramsType 3 - Euler DiagramsA Venn diagram show all possible logical relationships between a series of sets.But an Euler diagram only shows relationships that exist in real world.Syllogisms do not make sense logically…e.g. Apples are PizzasEuler diagrams do make sense logically…e.g. Apples and Pizzas are in one group, Paper and Pens are in another group. 101

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D Step 1: Separate them out into groups

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC DStep 1: Separate them out into groups Cakes Plants Glass Italian Food Trees Forests

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC DStep 1: Separate them out into groups Cakes Plants Glass Italian Food Trees Forests

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D Step 2: Make sure there are 6 items in each Venn

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D Step 2: Make sure there are 6 items in each Venn

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D Step 3: Compare category by category

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC DStep 3: Compare category by category Cakes Plants Glass Italian Food Trees Forests

Venn DiagramsExampleWhich of the following Venn diagrams correctly represents the is the relationship betweencakes, Italian Food, plants, trees, forests, and glass? ABC D Step 3: Compare category by category

Venn DiagramsExample 2Which of the following Venn diagrams correctly represents the is the relationship between ball-points, pens, rubbers and correction pens?

Venn to Text Type 4 112

Venn DiagramsType 4 - Venn to TextThey give you… a circular Venn Diagram (possibly with sections missing)They ask you… to choose the correct statement based on the Venn DiagramYou have to… use the information in the circular Venn to form conclusions 113

Venn DiagramsType 4 - Venn to TextThey give you… a circular Venn Diagram (possibly with sections missing)They ask you… to choose the correct statement based on the Venn DiagramYou have to… use the information in the circular Venn to form conclusions For these questions, it is best to look at the statements given and see which can be ruled out immediately or ones that take the least amount of time to rule out. Process of elimination is the way forward. 114

Venn DiagramsExampleThe following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire. Every homehas at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils. 115

Venn DiagramsThe following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire.Every home has at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils.First of all, we can see there are two missing values that are X and Y. It is a good idea to identify whatthese values represent:• X represents the number of gardens with daffodils, tulips, roses but not dandelions. 116• Y represents the number of gardens with tulips, roses but not daffodils and dandelions.

Venn Diagrams The following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire. Every home has at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils.Statement CLooking at these statements, the easiest to rule out is C because we just have to check if there is a region thathas all four circles overlapping or not. There is no such region, so zero gardens have all four flowers - C is false. 117

Venn DiagramsThe following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire.Every home has at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils.Statement BThe second easiest to rule out is B because we just have to add up the numbers that are not in overlappingregions. Adding up 4 + 3 + 7 + 6 = 20 - B is false. 118

Venn DiagramsThe following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire.Every home has at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils.Statement A Gardens with Roses Gardens with Tulips There are more gardens with roses than tulips. =9+X+Y+6+8+3 =2+3+6+8+X+Y = 26 + X + Y = 19 + X + Y 119

Venn DiagramsThe following diagram displays the number of flowers in the gardens of 100 citizens in Berkshire.Every home has at least one flower. Which of the following statements is true? A. There are more gardens that have tulips than roses. B. There are exactly 21 gardens with only one type of flower. C. The number of gardens with all four flowers can be calculated D. There are more gardens that have roses than daffodils.Statement D Gardens with Daffodils Gardens with Roses =4+2+9+X =9+X+Y+6+8+3 Statement D is correct = 15 + X = 26 + X + Y 120

Non Circular Venn to Text Type 5 121

Venn DiagramsType 5 - Non Circular Venn to TextThey give you… a non-circular Venn DiagramThey ask you… to choose the correct statement based on the Venn DiagramYou have to… use the information in the non-circular Venn to form conclusions 122

Venn DiagramsNon-circular Venn Diagrams The Venn Diagram below shows lunch options chosen by Medic Mind staff. How many more people have a meal with at least sweet potato pie than at least chicken breast? A. 6 B. 7 C. 8 D. 13

Venn DiagramsNon-circular Venn Diagrams The Venn Diagram below shows lunch options chosen by Medic Mind staff. How many more people have a meal with at least sweet potato pie than at least chicken breast? A. 6 B. 7 C. 8 D. 13 (7 + 6 + 5 + 2) - (2 + 5) = 13

Summary: Venn Diagrams 125

Venn DiagramsSummary• When you are presented with questions that do not already have a Venn diagram presented to you, draw your own!• Do not fall for the trap of just putting the numbers in the Venn without calculating them properly.• Read over the information as many times as you need.• If it is a Euler diagram question, try and think of as many possible conditions as possible. 126

Probabilistic Reasoning I Lesson 6 127

Probabilistic ReasoningIntroductionYou will be required to select the best possible response out of four statements regarding aprobability scenario.It will require you to use the basic principles of probability which we will now go overThe questions will be more likely to ask your reasoning behind a certain answer. This will meantthat you will have to work out the answer, and display the step by step working that you have used.

Probabilistic ReasoningBasic ProbabilityYou express a probability as either:• A decimal between 0 and 1• A percentage between 0 and 100%• A fraction between 0 and 10 means that an event is impossible. 1 means that an event is guaranteed.

Probabilistic ReasoningBasic ProbabilityProbability of an event happening is: Number of desired outcomes Total number of outcomes

Probabilistic ReasoningExample: Basic Probability Worked Example What is the probability of rolling a 3 on a normal dice?Number of desired outcomes - 1 (There is only one face with a 3 on the dice)Total number of outcomes - 6Probability = 1/6

Probabilistic ReasoningExample: Basic Probability Question Is it likely to be sunny for over three days over a two week period if the probability of it being sunny on any given day is 1/7? A.Yes, because it will be sunny for 7 days as 14 x 1/2 = 7 B.Yes, because it is more likely to be sunny than not sunny C.No, because it will be sunny for 2 days, as 14 x 1/7 = 2 D.No, because it is less likely to be sunny than not sunny

Probabilistic ReasoningExample: Basic Probability Question Is it likely to be sunny for over three days over a two week period if the probability of it being sunny on any given day is 1/7? A.Yes, because it will be sunny for 7 days as 14 x 1/2 = 7 B.Yes, because it is more likely to be sunny than not sunny C.No, because it will be sunny for 2 days, as 14 x 1/7 = 2 D.No, because it is less likely to be sunny than not sunny

Probabilistic ReasoningExample: Basic Probability Expected Frequency = Probability x Number of Repeats Probability is 1/7. The number of repeats is the number of days, which is 14. Expected Frequency = 14 x 1/7 = 2 days

Probabilistic ReasoningRepeat of Same EventsThis is when the same event occurs three times and the probability of each event doesnot change one after the other. Probability of A three times = Probability of A x Probability of A x Probability of A

Probabilistic ReasoningExample: Repeat of same events Question There are four marbles in a bag. 3 of which are green and one of which is red. What is the probability of picking a red marble from a bag three times if the marble is replaced in the bag after each selection? A.0 B.1 / 16 C.1 / 64 D.1 / 256

Probabilistic ReasoningExample: Repeat of same events Question There are four marbles in a bag. 3 of which are green and one of which is red. What is the probability of picking a red marble from a bag three times if the marble is replaced in the bag after each selection? A.0 B.1 / 16 C.1 / 64 D.1 / 256

Probabilistic ReasoningExample: Repeat of same events Solution Remember, because the marbles are replaced each event is independent. Probability of red three times = Probability of red x Probability of red x Probability of red Probability of red three times = 1/4 x 1/4 x 1/4 = 1/64

Probabilistic ReasoningDifferent events that are mutually exclusiveWhen there are two different events, and the probability of one event does not affect thechance of the other event happens. Probability of A and B = Probability of A x Probability of B

Probabilistic ReasoningExample: Different events that are mutually exclusive Question The probability of England winning the World Cup is 1/24. Olivier Giroud plays three matches. In each individual match, the probability of him scoring is 1/5. Is the probability of Olivier Giroud scoring in every match greater than the probability of England winning and Giroud scoring in at least one match. A. Yes, because the probability of England winning the World Cup and Olivier Giroud scoring is 1/144, and the probability of Giroud scoring in every match is 1/216. B. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 1 / 144, and the probability of Giroud scoring in every match is 1/216. C. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 61/3000, and the probability of Giroud scoring in every match is 1/125. D. No, because the probability of England winning the World Cup and Olivier Giroud scoring is equal to the probability of Giroud scoring in every match.

Probabilistic ReasoningExample: Different events that are mutually exclusive Question The probability of England winning the World Cup is 1/24. Olivier Giroud plays three matches. In each individual match, the probability of him scoring is 1/5. Is the probability of Olivier Giroud scoring in every match greater than the probability of England winning and Giroud scoring in at least one match. A. Yes, because the probability of England winning the World Cup and Olivier Giroud scoring is 1/144, and the probability of Giroud scoring in every match is 1/216. B. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 1 / 144, and the probability of Giroud scoring in every match is 1/216. C. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 61/3000, and the probability of Giroud scoring in every match is 1/125. D. No, because the probability of England winning the World Cup and Olivier Giroud scoring is equal to the probability of Giroud scoring in every match.

Probabilistic ReasoningExample: Repeat of same events QuestionThe probability of England winning the World Cup is 1/24. Olivier Giroud Probability of England Winning and Giroud Scoring Atplays three matches. In each individual match, the probability of him scoring Least Onceis 1/5.Is the probability of Olivier Giroud scoring in every match greater than the First work out the probability of Giroud scoring at least once:probability of England winning and Giroud scoring in at least one match. = 1 - probability of him not scoring in any matchA. Yes, because the probability of England winning the World Cup and Olivier = 1 - (4/5 x 4/5 x 4/5) Giroud scoring is 1/144, and the probability of Giroud scoring in every match is = 1 - 64/125 1/216. = 61/125B. No, because the probability of England winning the World Cup and Olivier We need to multiply the probability of England winning by the Giroud scoring is 1 / 144, and the probability of Giroud scoring in every match is probability of Giroud scoring at least once: 1/216. 1/24 x 61/125 = 61/3000 = 2.03% chanceC. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 61/3000, and the probability of Giroud scoring in every match is 1/125.D. No, because the probability of England winning the World Cup and Olivier Giroud scoring is equal to the probability of Giroud scoring in every match.

Probabilistic ReasoningExample: Repeat of same events QuestionThe probability of England winning the World Cup is 1/24. Olivier Giroud Probability of Giroud in Every Matchplays three matches. In each individual match, the probability of him scoringis 1/5. Probability of A x A x AIs the probability of Olivier Giroud scoring in every match greater than the 1/5 x 1/5 x 1/5 = 1/125 = 0.8% chanceprobability of England winning and Giroud scoring in at least one match. The probability of Giroud scoring three times is lower than theA. Yes, because the probability of England winning the World Cup and Olivier probability of England winning the World Cup and Olivier Giroud Giroud scoring is 1/144, and the probability of Giroud scoring in every match is scoring a goal. 1/216.B. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 1 / 144, and the probability of Giroud scoring in every match is 1/216.C. No, because the probability of England winning the World Cup and Olivier Giroud scoring is 61/3000, and the probability of Giroud scoring in every match is 1/125.D. No, because the probability of England winning the World Cup and Olivier Giroud scoring is equal to the probability of Giroud scoring in every match.

Probabilistic Reasoning II Lesson 7 144

Probabilistic ReasoningEither One of Two Events (where both cannot happen)Sometimes you get questions where they ask you to work out the probability of either A or Bhappening, but both are not possible. For example, if you roll a dice, the probability ofgetting 1 or 2 is the sum of each individual probability. Probability of A or B = Probability of A + Probability of B

Probabilistic ReasoningExample: Either One of Two Events (where both cannot happen) Probability of A or B = Probability of A + Probability of BWhat is the probability of rolling a 3 or 4 on a normal dice?

Probabilistic ReasoningExample: Either One of Two Events (where both cannot happen)What is the probability of rolling a 3 or 4 on a normal dice?Probability of 3 or 4 = Probability of rolling 3 + Probability of rolling 4Probability of 3 or 4 = 1/6 + 1/6 = 1/3

Probabilistic ReasoningEither One of Two Events (where both can happen)If you had separate events which could both happen, the previous formula does notwork because you need to factor in the possibility that both could happen. Probability of A or B = 1 - Probability of neither happening

Probabilistic ReasoningExample: Either One of Two Events (where both can happen) Worked Example The chance of Khaled wearing a flowery shirt is 1/4, and the chance of Yogi wearing a black t- shirt is 1/3. What is the probability of at least one happening?

Probabilistic ReasoningExample: Either One of Two Events (where both can happen) Worked Example The chance of Khaled wearing a flowery shirt is 1/4, and the chance of Yogi wearing a black t- shirt is 1/3. What is the probability of at least one happening? To work out the probability of either happening you need to add: • The probability of just Khaled wearing a flowery shirt • The probability of just Yogi wearing a black t-shirt • The probability of both Khaled wearing a flowery shirt and Yogi a black t-shirt.