MAS113 Introduction To Probability And Statistics

2y ago
197 Views
9 Downloads
861.85 KB
105 Pages
Last View : 1d ago
Last Download : 2m ago
Upload by : Laura Ramon
Transcription

IntroductionSet theory and probabilityMeasureMAS113 Introduction to Probability andStatisticsDr Jonathan JordanSchool of Mathematics and Statistics, University of Sheffield2020–21Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureStudying probability theoryThere are (at least) two ways to think about the study ofprobability theory.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasurePure mathematical approachProbability theory is a branch of Pure Mathematics.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasurePure mathematical approachProbability theory is a branch of Pure Mathematics.We start with some axioms and investigate the implications ofthose axioms; what results are we able to prove?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasurePure mathematical approachProbability theory is a branch of Pure Mathematics.We start with some axioms and investigate the implications ofthose axioms; what results are we able to prove?Probability theory is a subject in its own right, that we canstudy (and appreciate!) without concerning ourselves with anypossible applications.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasurePure mathematical approachProbability theory is a branch of Pure Mathematics.We start with some axioms and investigate the implications ofthose axioms; what results are we able to prove?Probability theory is a subject in its own right, that we canstudy (and appreciate!) without concerning ourselves with anypossible applications.We can also observe links to other areas of mathematics.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureModelling approachProbability theory gives us a mathematical model forexplaining chance phenomena that we observe in the realworld.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureModelling approachProbability theory gives us a mathematical model forexplaining chance phenomena that we observe in the realworld.As with any mathematical theory, it may not describe realityperfectly, but it can still be extremely useful.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureModelling approachProbability theory gives us a mathematical model forexplaining chance phenomena that we observe in the realworld.As with any mathematical theory, it may not describe realityperfectly, but it can still be extremely useful.We must think carefully about when and how the theory canbe applied in practice.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureThis courseWe will be studying probability from both perspectives.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureThis courseWe will be studying probability from both perspectives.Applications of probability theory are important andinteresting, but as a mathematician, you should also developan understanding and an appreciation of the theory in its ownright.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivationWe are often faced with making decisions in the presence ofuncertainty.A doctor is treating a patient, and is considering whetherto prescribe a particular drug. How likely is it that thedrug will cure the patient?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivationWe are often faced with making decisions in the presence ofuncertainty.A doctor is treating a patient, and is considering whetherto prescribe a particular drug. How likely is it that thedrug will cure the patient?A probation board is considering whether to release aprisoner early on parole. How likely is the prisoner tore-offend?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivationWe are often faced with making decisions in the presence ofuncertainty.A doctor is treating a patient, and is considering whetherto prescribe a particular drug. How likely is it that thedrug will cure the patient?A probation board is considering whether to release aprisoner early on parole. How likely is the prisoner tore-offend?A bank is considering whether to approve a loan. Howlikely is it that the loan will be repaid?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivationWe are often faced with making decisions in the presence ofuncertainty.A doctor is treating a patient, and is considering whetherto prescribe a particular drug. How likely is it that thedrug will cure the patient?A probation board is considering whether to release aprisoner early on parole. How likely is the prisoner tore-offend?A bank is considering whether to approve a loan. Howlikely is it that the loan will be repaid?A government is considering a CO2 emissions target.What would the effect of a 20% cut in emissions be onglobal mean temperatures in 20 years’ time?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureDescribing uncertaintyWe often use verbal expressions of uncertainty (“possible”,“quite unlikely”, “very likely”), but these are often inadequateif we want to communicate with each other about uncertainty.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureDescribing uncertaintyWe often use verbal expressions of uncertainty (“possible”,“quite unlikely”, “very likely”), but these are often inadequateif we want to communicate with each other about uncertainty.Consider the following example.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureDisease testYou are being screened for a particular disease. You are toldthat the disease is “quite rare”, and that the screening test isaccurate but not perfect.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureDisease testYou are being screened for a particular disease. You are toldthat the disease is “quite rare”, and that the screening test isaccurate but not perfect.If you have the disease, the test will “almost certainly” detectit, but if you don’t have the disease, there is “a small chance”the test will mistakenly report that you have it anyway.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureDisease testYou are being screened for a particular disease. You are toldthat the disease is “quite rare”, and that the screening test isaccurate but not perfect.If you have the disease, the test will “almost certainly” detectit, but if you don’t have the disease, there is “a small chance”the test will mistakenly report that you have it anyway.The test result is positive. How certain are you that you reallyhave the disease?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureQuantifying uncertaintyClearly, in this example and in many others, it would be usefulif we could quantify our uncertainty.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureQuantifying uncertaintyClearly, in this example and in many others, it would be usefulif we could quantify our uncertainty.In other words, we would like to measure how likely it is thatsomething will happen, or how likely it is that some statementabout the world turns out to be true.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureQuantifying uncertainty continuedWe will introduce a theory for measuring uncertainty:probability theory.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureQuantifying uncertainty continuedWe will introduce a theory for measuring uncertainty:probability theory.We have two ingredients: set theory, which we will use todescribe what sort of things we want to measure ouruncertainty about, and, appropriately, measure theory, whichsets out some basic rules for how to measure things in asensible way.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivation - the need for set theory and measuresIf you have studied probability at GCSE or A-level, you mayhave seen a definition of probability like this:Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivation - the need for set theory and measuresIf you have studied probability at GCSE or A-level, you mayhave seen a definition of probability like this:Suppose all the outcomes in an experiment are equally likely.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureMotivation - the need for set theory and measuresIf you have studied probability at GCSE or A-level, you mayhave seen a definition of probability like this:Suppose all the outcomes in an experiment are equally likely.The probability of an event A is defined to beP(A) : number of outcomes in which A occurs.total number of possible outcomesDr Jonathan Jordan(1)MAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureExampleExampleA deck of 52 playing cards is shuffled thoroughly. What is theprobability that the top card is an ace?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureExampleExampleA deck of 52 playing cards is shuffled thoroughly. What is theprobability that the top card is an ace?The ‘definition’ can work for examples such as this (with afinite number of equally likely outcomes), but we soon run intodifficulties:Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureHarder exampleExampleImagine generating a random angle between 0 and 2π radians,where any angle is equally likely.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureHarder exampleExampleImagine generating a random angle between 0 and 2π radians,where any angle is equally likely.(For example, consider spinning a roulette wheel, andmeasuring the angle between the horizontal axis (viewing thewheel from above) and a line from the centre of the wheelthat bisects the zero on the wheel).Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureHarder exampleExampleImagine generating a random angle between 0 and 2π radians,where any angle is equally likely.(For example, consider spinning a roulette wheel, andmeasuring the angle between the horizontal axis (viewing thewheel from above) and a line from the centre of the wheelthat bisects the zero on the wheel). What is the probability that the angle is between 0 and 2radians?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureHarder example continued You might guess that the answer should bedefinition in (1) causes us problems!Dr Jonathan Jordan2,2πbut theMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureHarder example continued You might guess that the answer should bedefinition in (1) causes us problems!2,2πbut theThere are infinitely many possible angles that could begenerated,and there are infinitely many angles between 0 and 2 radians. Clearly,we can’t divide infinity by infinity and come up with 2π2 .Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureAre outcomes equally likelyIt may not make much sense to think of outcomes as equallylikely:Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureAre outcomes equally likelyIt may not make much sense to think of outcomes as equallylikely:ExampleWhat is the probability that a team from Manchester will winthe Premier League this season?Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureAre outcomes equally likelyIt may not make much sense to think of outcomes as equallylikely:ExampleWhat is the probability that a team from Manchester will winthe Premier League this season?We will resolve this by constructing a more general definitionof probability, using the tools of set theory and a special typeof function called a measure, and we’ll show how choosingdifferent types of measure will give us sensible answers in boththe above examples.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet theory and probabilitySet theory is covered in more detail in MAS110; in this modulewe consider set theory in the context of probability.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet theory and probabilitySet theory is covered in more detail in MAS110; in this modulewe consider set theory in the context of probability.We consider uncertainty in the context of an experiment.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet theory and probabilitySet theory is covered in more detail in MAS110; in this modulewe consider set theory in the context of probability.We consider uncertainty in the context of an experiment.(Here we use the word experiment in a loose sense to meanobserving something in the future, or discovering the truestatus of something that we are currently uncertain about.)Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSample spaceIn an experiment, suppose we want to consider how likelysome particular outcome is.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSample spaceIn an experiment, suppose we want to consider how likelysome particular outcome is.We might start by considering what all the possible outcomesof the experiment are. We can use a set to list all the possibleoutcomes.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSample spaceIn an experiment, suppose we want to consider how likelysome particular outcome is.We might start by considering what all the possible outcomesof the experiment are. We can use a set to list all the possibleoutcomes.DefinitionA sample space is a set which lists all the possible outcomesof an ‘experiment’.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureUniversal setIn set theory, we sometimes work with a universal set S,which lists all the elements we wish to consider for thesituation at hand.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureUniversal setIn set theory, we sometimes work with a universal set S,which lists all the elements we wish to consider for thesituation at hand.(Note that, in spite of the word “universal”, the choice of Sdepends on context.)Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureUniversal setIn set theory, we sometimes work with a universal set S,which lists all the elements we wish to consider for thesituation at hand.(Note that, in spite of the word “universal”, the choice of Sdepends on context.)In the context of probability, the sample space will play therole of the universal set S.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureExamplesExampleExamples of sample spacesDr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureEventsDefinitionAn event is a subset of a sample space.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureEventsDefinitionAn event is a subset of a sample space.If the outcome of the experiment is a member of the event, wesay that the event has occurred.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureEventsDefinitionAn event is a subset of a sample space.If the outcome of the experiment is a member of the event, wesay that the event has occurred.ExampleExamples of eventsDr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet operations and eventsGiven a sample space S and two subsets/events A and B, theset operations union, intersection, complement and differenceall define further events.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureUnionThe union A B corresponds to either A occurring or to Boccuring (or to both occurring).Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureUnionThe union A B corresponds to either A occurring or to Boccuring (or to both occurring).We can visualise this using a Venn diagram. The rectanglerepresents the universal set, the two circles represent thesubsets A and B, and the shaded area represents the set A B.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureIntersectionThe intersection A B corresponds to both A and Boccurring.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureEmpty setIf there are no elements in S that are both in A and in B, thenA B .Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureEmpty setIf there are no elements in S that are both in A and in B, thenA B .In this case we say that A and B are mutually exclusive.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureComplementThe complement of A, A, corresponds to A not occurring.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureComplementThe complement of A, A, corresponds to A not occurring.(Alternative notation: Ā is also written as AC and A0 .)Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureComplementThe complement of A, A, corresponds to A not occurring.(Alternative notation: Ā is also written as AC and A0 .)Note that S̄, and that the event is one which cannothappen, as it contains no elements.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet differenceThe set difference A \ B corresponds to A occurring but Bnot.Dr Jonathan JordanMAS113 Introduction to Probability and Statistics

IntroductionSet theory and probabilityMeasureSet differenceThe set difference A

Dr Jonathan Jordan MAS113 Introduction to Probability and Statistics. Introduction Set theory and probability Measure Motivation - the need for set theory and measures If you have studied probability at GCSE or A-level, you may have seen a de nition of probability like this:

Related Documents:

Joint Probability P(A\B) or P(A;B) { Probability of Aand B. Marginal (Unconditional) Probability P( A) { Probability of . Conditional Probability P (Aj B) A;B) P ) { Probability of A, given that Boccurred. Conditional Probability is Probability P(AjB) is a probability function for any xed B. Any

Pros and cons Option A: - 80% probability of cure - 2% probability of serious adverse event . Option B: - 90% probability of cure - 5% probability of serious adverse event . Option C: - 98% probability of cure - 1% probability of treatment-related death - 1% probability of minor adverse event . 5

Chapter 4: Probability and Counting Rules 4.1 – Sample Spaces and Probability Classical Probability Complementary events Empirical probability Law of large numbers Subjective probability 4.2 – The Addition Rules of Probability 4.3 – The Multiplication Rules and Conditional P

Probability measures how likely something is to happen. An event that is certain to happen has a probability of 1. An event that is impossible has a probability of 0. An event that has an even or equal chance of occurring has a probability of 1 2 or 50%. Chance and probability – ordering events impossible unlikely

Engineering Formula Sheet Probability Conditional Probability Binomial Probability (order doesn’t matter) P k ( binomial probability of k successes in n trials p probability of a success –p probability of failure k number of successes n number of trials Independent Events P (A and B and C) P A P B P C

Target 4: Calculate the probability of overlapping and disjoint events (mutually exclusive events Subtraction Rule The probability of an event not occurring is 1 minus the probability that it does occur P(not A) 1 – P(A) Example 1: Find the probability of an event not occurring The pr

Solution for exercise 1.4.9 in Pitman Question a) In scheme Aall 1000 students have the same probability (1 1000) of being chosen. In scheme Bthe probability of being chosen depends on the school. A student from the rst school will be chosen with probability 1 300, from the second with probability 1 1200, and from the third with probability 1 1500

or a small group of countries, we explore possible drivers behind the decline in income inequality in Latin America as a whole. To undertake this task, we utilize an array of methodologies—including correlation and econometric techniques. To start, we look at simple correlations between changes in policy variables and changes in income inequality