 But please take a note of This allocation. This is where you're supposed to be in the tutorials. Okay Now, you know and if you have any questions Then Vicky is sitting right there and she will be happy to clarify everything in the break now due to all this information we had in the last introductory Lecture to this course there was a little information which we missed out on and I Will introduce this in information today it concerns a Little more specific detail on the definition of risk and then also a little more information regarding motivation For risk assessments and then we will go in and have a look at the probability theory We will talk about different interpretations of probability Then I will introduce the concept of sample space and events Which can take place in sample space Then we will have the three axioms of probability theory you can consider The axioms of probability theory to be the rules of the game which we are playing here and Then in the end of the lecture, we will talk about two very important concepts Namely conditional probability and the theorem of base Are you ready? Risk is the word which you probably all know and as we discussed briefly at the last lecture There are many different understandings of what is risk really But here of course we have a rather specific Idea about what risk really is and we will define this now So risk is the characteristic of an activity. So an activity is a relatively abstract notion and risk relates to all possible events of Which there are in E Which may follow as the result of the activity. So we are dealing with an activity and This activity is associated with in E possible events and the risk contribution From each of those individual events EI so now we have EI I going from one up to N. E The risk contribution for each of those events is defined through the product between the probability that the event EI will take place and The consequences shoot the event EI take place so the events of Which we have in E can be described by their probability of occurrence and Their associated consequences and for each of those events We calculate the risk contribution as a product of the probability that the event will take place and The consequences that the event will take place now because we have in E events associated With the activity we are considering then we have to sum up or each of those individual risk contributions and This is why we have this summation sign here and that now Constitutes the total risk associated with the activity a Okay, so that's pretty straightforward now I will just tell you that the assessed risk associated with some activity which we may want to undertake or evaluate the feasibility of as engineers is the one of the major building stones for decision-making so we Evaluate the risk for the different decision alternatives we have and then we compare the risk and in that way We can rank the different Risk alternatives we have available or say decision alternatives we have available. Okay So therefore the concept of risk is very important and you see that We are dealing with two properties here, which we need to to be concerned about namely the probability that events will take place and their consequences and in this lecture so Statistics and probability you will learn all the basics you need in order to Assess the probabilities of events which are of interest Activities I introduced Activity is relatively abstract notion activities can be anything it can be a Construction procedure in order to construct some type of infrastructure It can be the design of the house it can be the development of Power supply systems or whatever all those can be considered to be Activities which we have to evaluate as engineers Now uncertainties have to be considered in the decision-making throughout all phases of the life of an Indian Yang activity and if you look at this Very nice bridge. Does anybody recognize this bridge? I'm sure you do come on Have anybody been in France? How many French people do we have here? Ah, okay, so it's the Ponte de Normandie bridge Aha It's concepted by a very famous French engineer It's a very nice bridge and when we are dealing with engineering activities like the conception and construction The operation of bridges like that. We really have to Look at the whole life cycle of such a Bridge starting with the idea and the concept Going over to the planning and the feasibility studies After the planning and feasibility studies We go into what we call investigations and and tests so you have to imagine that Some people are sitting behind a desk at some point in time and they get the idea Well, maybe we should build a bridge here in the planning and the feasibility studies they are evaluating different options for for Constructing and and developing a bridge project They have different options and they want to consider which option is the better one then when they have maybe narrowed down the scope a little and they are Considering only two or three Different bridge options, then they go into investigations and tests. So investigations and tests Include going on the site Where the bridge is supposed to be and then you can Investigate which are the more precise properties of the soil what time What type of environment do we have where we want to locate the bridge in terms of of wind and temperature variations? You can also make seismic Assessments and when we are of course when we are dealing with the bridge then also the environment in regard to currents in the sea and waves Needs to be assessed From the investigation and and the testing phase of such a project. We go over to the phase of the design So this is a typical disk top engineering activity where engineers are now sitting and they are trying to find out What should be the dimensions of the pylons? how What should be the resistance of the roadway lanes? Etc also they choose materials they try to to identify how to build The bridge but everything still on paper Then it goes over to the manufacturing phase. So in this phase based on the on the design The components and the materials for the bridge will be will be built in factories sometimes All over the world and then they will be brought together in the next phase where the bridge project is executed that means the real construction process on site and After the execution phase we go over to the operation and maintenance phase now Typically I Would say that the idea and Concepts plus the planning and the feasibility studies could be Taking place within a period as a ranging between two to many years Many many years sometimes when we are looking for instance at a project like the machine bridge project Which I'm sure you have all heard about This is a typical project where Where the idea conception planning and feasibility study phase has been extremely long? Most major engineering companies they have several projects lying in there in their drawers Ready to be applied if at some point in time Italy really would like to build the machine bridge So this phase has been very long now the investigation testing design manufacturing execution Is something which is typically limit to? Depending on the size of the project may be four to six years But then we come into the operation and the maintenance phase and this phase Should be really long. That's the whole idea of developing such a project. You want to have a bridge Which is lasting for a long time fulfilling its purpose facilitating traffic from point A to B Now after the face of the operation and maintenance This is the face where the bridge is being used and Kept in good condition then at some point in time when the bridge Becomes obsolete or the activity which we're dealing with becomes obsolete Then we have to end the activity and when we are dealing with The structures like bridges tunnels at some point in time We have to take them out of service. We have to decommission them and That in many cases means that we have to remove them so we have to break them down and To the extent possible. We also have to recycle the materials and in many cases Of course, we are also leaving when we are removing a structure like like this or tunnel we are We are maybe leaving some sort of scar in the landscape Some traces in the environment and then we have to clean that up. Okay, and this this is An activity. This is a phase of engineering projects, which has been up until recently has been greatly omitted Typically when Projects are being planned, but within I would say within the last five to ten years the decommissioning has been realized to be a very very important phase of The life of such activities So please remember that and When we are dealing with those phases, of course engineers are involved in all these phases of projects and The role of the engineer is to make sure that The activities which go on in these different phases that they are economically feasible So we have to be economically efficient. We have to use the resources of the society in the most efficient manner as That takes innovative and Good engineering in order to be able to do that at the same time. We have to safeguard personnel people and We have to safeguard the environment so these are the boundary conditions we have to consider and You can imagine that when we are dealing with economical feasibility we are dealing with safety for the persons involved or exposed and we are dealing with the environment then There's a certain in many ways that contradictions So the more we invest the more expensive will our activity our project be the more we invest However, also the more safety we can achieve for persons and the environment So that's on the one side, but that is associated with high cost in General we would like to reduce cost, but as we reduce cost then also typically at some point in time We are reducing safety for Personnel and we are reducing the safety for the qualities of the environment. So this is a weighing and In this weighing we have to consider what is What do we consider as a society to to be? Acceptable risks. Okay, so when we're dealing with safety of personnel and environment the Attributes we are concerned about are the risks Because there's a lot of uncertainties associated with all these phases of an engineering activity There's a lot. We don't know and The amount of knowledge we have Basically depends on in which phase of a project We are now just to mention some of the uncertainties Which could be relevant Let's say in in a planning and feasibility Phase of such a project. We are dealing with For instance uncertainties associated with the traffic volume, which will be Transferred on the bridge. We don't know we have some models which can describe the anticipated traffic volume But we don't know and the feasibility of the whole project strongly depends on the amount of traffic Which is in the future will cross the bridge So that is is a major player Now also the loads which will be acting on on the bridge. They're also associated associated with significant uncertainty We may have models describing the variations in the wind the Variability of waves and currents We may have models indicating What type of earthquakes we can expect in such a region? Etc. But they are only models We don't know for sure and there for that reason the loads are uncertain and we have to take that into account now the same also applies for the resistance of the individual components and The structural system as a whole for the bridge again. We have models We can make all sorts of calculations But in the end models are only models and there is a certain variability in the performance of the materials Which are used to construct such bridges also the soil on which the bridge is constructed We are able to drill holes and we can take samples of the soil and we can use these samples to analyze the Mechanical properties of the soil, but again, we can only use these Informations in the framework of engineering models and the models are uncertain. So in the end Also the resistances are associated with significant Uncertainties and then comes the problem of degradation. So what is going to happen with the materials in the future? The only thing which we can be sure of is that the materials will degrade Now the rate of degradation depends on what we do in order to avoid the degradation So our choice of material relative to the environment where we are locating a bridge like this has a very big significance But again corrosion fatigue are phenomena we have to deal with and we don't understand these phenomena very well Again, we have some engineering models in some cases under some conditions the models are working very nicely Especially when we're dealing with laboratory conditions in those cases our models are especially Nice, but when we move out into the real world we have problems with the models and for this reason any prediction of future degradation is associated with significant uncertainty and And that also has the effect of what we call the service life the service life of such an activity or a structure Is in general a very subjective concept we can come up with any set of definitions of service life The when we're dealing here in the planning of And and feasibility study phase we also of course when we we want to identify a good concept What we would really like to know also is what is the price of? Developing the individual optional Concepts so the manufacturing cost the execution costs, but also the decommissioning cost So how much is it going to take the structure away again when? after hopefully many years the Project the bridge becomes obsolete So uncertainties are extremely important, but based so what we can do as engineers based on these uncertainties is that we can we can model these uncertainties probabilistically and when we have developed such probabilistic models then we can Identify the events which can which can result in in economical losses Which can result in damages to the qualities of the environment and which can result in? loss of lives or say injuries Based on these probabilistic models Of the uncertainties we can calculate the probabilities that such events will take place and in by doing that By evaluating the associated risks we have a means for deciding on what are the best options in each of those different phases of The lifetime of an engineering activity, and this is called risk management and this is where Statistics and probability fits in to engineering. I'll show you One little example here. This is a project. I was involved in in the end I say mid 90s to the end of the 90s What you see here are some offshore structures, which are located in the well They are located in the North Sea This broken line you see here is the it's a border between Norway and and The United Kingdom So we're dealing with an interesting phenomena. We have an oil field Which apparently is located just down below the border and and at some point in time an oil company Which at that time? Had the name Elf ELF French operator Concepted and and had these platforms here constructed Of course for exploiting the oil and gas reserves, which are located here under the in the soil on this In this location now You see three different You see three steel platforms these platforms here are steel platforms and You see three concrete platforms It's not very clear on this drawing here, but the concrete platforms are all different the steel platforms are more or less the same type The structures which we are focusing on now are called TP1 TCP2 and CDP1 If you look at the TCP2 This is a typical Platform which we call a Condip platform type. It's a gravity-based platform. So it's based. So it's built up with these concrete cells And then we have three legs also made of concrete and on top of these three legs you then place the the the factory Where the oil is is treated and the gas is treated before it's it's sent into some Pipelines and then it's brought on shore These things are relatively big the the water depths at least this location is around I'd say close to 100 meters The height of these concrete cylinders is around 50 meters. We're dealing with Big structures really really big structures if you would put these structures next to This building here then this building would look ridiculously small So you might also imagine that they are pretty heavy these things This here is Is called a sea tank structure? Well, it's similar to the other one But here you you see that the profile is a square profile Again the concept is the same we have concrete cylinders and then we have two legs in this case on which the The process facilities are located then we have the last one which is a concept developed also by the French company called Dory they are Experts in in develop developing Structures like this Mostly operating offshore This is a Doritang tank. It's also it's it's constructed of of concrete, but In comparison to the other two this this thing here Is built of pre-fabricated components? That's quite interesting. So you make all the components You sail them out and you construct by putting the components together the other structures here They are constructed in a very interesting way You start by constructing the very bottom of these platforms onshore On a location very close to the shoreline then when you have you have constructed the very lower part of This tank you can imagine that you have like a big boat and then you you fill in water around it and then it floats Then now you sail it out into a fjord or a safe location on the sea and then you Start continuing building up the walls and of course as the as a boat gets heavier and heavier It also sinks because you're always constructing concrete So you are basically you're building a boat which gets Larger and larger, but basically below the the the surface and You continue in that way and in the end in the end basically you have you have the Only the shafts Sticking out of the water you have this big big structure floating below the water. It's not touching the bottom You're still constructing and then You have this strange Sheep subsea ship with only the shafts is digging up of the water When when you are in that situation And and you you can you can control that by pumping also water into the structure because of course It has a lot of buoyancy at some point in time then you pump out the water again and then the whole structure floats up on the surface and then you pull it out on the location in the ocean where you want to locate the the the platform Then you pump water into the tanks again and you sink it On the location and you place the the top sides on on the structure And then you have a platform Now these are very significant offshore operations, but With these platforms designed in in this period of time It was this is that mainly let's say the period of time where the oil and gas resources in the North Sea Were started to be exploited at those early times There was not so much focus on the environment And for that reason the decommissioning phase was not seriously considered when structures like these were designed and However, when there was no more oil and gas on this location the Norwegian government Of course was concerned about what would happen now with these structures the concrete structures Especially the steel structures are relatively easy to get away. You just cut Cut the structures off here down at the bottom. They are Founded on some piles you cut them off and then you can you can float them up You can put them on some some ships and you can sail them away That's in general not so complicated, but these concrete structures. They're really big and they're really heavy and They the concern was that they needed to somehow Decommission them and there are different options on how to decommission and what should they do? Now the weights for each structure is around 250,000 tons and the estimated cost when I got involved in this project was Somewhere between 200 and 600 million Swiss francs for the removal of each of those structures This is a wide range, but it's just reflects that such numbers are very uncertain Depending on what happens Now the operator Would like to evaluate options for removing these structures Taking into account of course the safety of personnel involved In all the processes of the removal also the safety of the environment and and of course also They were concerned about the costs And another aspect was that during such the decommissioning operation and in the preparation of the decommissioning and the identification of the different possibilities for removing the structures It was a major concern to communicate with different interest groups different interest interest groups are among others Greenpeace Also the fissures which are fishing in in the area where the platforms are located are of course important the general public Is also an important interest group? Imo is the international maritime organization and they are regulating activities on sea So of course, they were also an important interest group In general three options were considered The first one is called refloat and demolition onshore in the refloat What you do is that you pump out the water and then by various means you also Establish an over pressure Between the bottom of the platform and the seabed such that you can you can Overcome the suction Between the platform and the seabed and then you you you basically you remove it in the reverse way Which you originally installed it? So you get it up to the surface and then you can tow it away. You can either demolish it onshore or offshore Some one of the options was simply to rebuild everything So every piece of concrete every single piece of steel was Recycled the concrete where it was planned to be used in roadway works and And the steel of course to be melted and used. I don't know maybe for bicycles or cars Another option was demolition offshore And demolition offshore there again various options you can more or less recycle, but everything Offshore you can also just clean the structure completely take off everything Which is relatively easy to take off and then you can you can sail the structure on Location where you have very deep water and then you can pull out the plug and you can say goodbye structure Now you're gone And then there's an option which According to IMO the international maritime organization is required as a minimum And that is to make sure that there's a free passage of 55 meter water depth So everything Between the surface and down to a depth of 55 meters it has to be free passage according to these regulations and That could be achieved by cutting off the legs just over the tanks So these these were three options and I was involved in in assessing the risks associated with these three decommissioning options So what we did is that we tried to identify the different hazard scenarios Chronologically, so we tried to identify the different time phases in these three different removal or decommissioning options What would take place in different time steps during these three different? options for removal and What could go wrong? Okay and Then we identified all events which we assumed to be associated with consequences and We quantified the probabilities that these events would take place and We used a certain type of of tool for doing these analysis for doing these risk assessments and they are called Bayesian Nets and If you come back to my lectures in the seventh semester, you will learn much more about what that is basically Bayesian Nets is is a probabilistic analysis tool where you can you can model variables which are all Which are involved in the problem and you can you can model the uncertainties Which are associated with these variables now? I'll just illustrate One of the nets we would be dealing with hundreds of these different nets Which are relevant to assess the risks in the different individual faces for the removal For the different options now here you see one cell which is called cell integrity That's that is the tightness of say the water tightness of one of these concrete cells in the Kondi platform And the cell integrity Can be Can be there so either it's okay or? either it's it's leaking and One of the problems with this particular type of platform is that between the cells There are some hollow spaces You can you can see it's indicated here in the figure you can imagine that there is a star-shaped cell going down All the way to the bottom between the cells and and this this space is connected directly to the sea so you have inside the cell When you have no air No, when when all the water is removed in the cell then you have you have air in the cell and then Outside the cell down at the bottom. You have the enormous the enormous pressure due to the to the water And you have an enormous pressure difference. So inside the cell there's air Just outside the cell there's water at a water depth of 100 meters and that gives enormous stresses in the concrete structure and these stresses were not Considered when the structure was designed and of course if you have a cell Breaking in in in the face of trying to lift off the platform of the of the bottom The cell is fracturing then the whole mission is failing and that would be a major Cost problem and also a safety problem for the people involved now. I want to show you as a small thing Let's see if that works. No But never give up So I can show it here this basically illustrates what's going on Yeah, you see these These the empty spaces open spaces due to the pressure difference this phenomenon here can take place and if that takes place Air going out and we have of course water going into the tank Then we have the platform going down again and in this situation the structure is The structure is no longer in a condition where it can be removed and the whole mission has failed with a lot of economical resources having been invested and even worse if that Happens then there are no options for removing the platform afterwards because then it's absolutely unsafe to work with the platform and That would be very bad. It would be very bad for the for the environment the interest groups would not be happy and So this was a major concern So based on our risk assessments one of the results which we were able to derive is that as a function of time Now you may imagine that these time steps here They go over several years Where the removal operations are being planned? Tests and investigations are ongoing and at some point in time also the the Different activities offshore are being conducted and What we can evaluate from the very beginning from the very planning phase up until and during the execution of the removal Is that we can calculate the probability of mission success? So what is the probability that our? Concept will be successful this probability is is what we have here. This is a black line and No What have we here? We have the expected cost Going going up. This is a black line And then we have the probability of mission failure going down, okay? This is a red one not mission success, but mission failure and the the very One of the various was how much money for one of these concepts for Removal how much money was and is required to invest before we have a sufficient Low probability of mission failure yeah, and These curves could be developed for all different types of concepts for removing the platform And that was a part of the basis for selecting in what way to remove the platforms So that was just to give you a small example of how risk assessments and and The treatment of uncertainties in engineering decision-making is also being used in practice now we are also Dealing with other activities like structural design and when we are dealing with with projects and structures which are unusual Exceptional then these are situations where there are no there are no rules. There are no guidelines in general which can be used Directly for engineering these projects In those situations we have as engineers we have to develop the basis for the development of projects ourselves So there's no code which can which can describe how to design and construct an exceptional Structure one example here is the great built bridge Which is the here under construction. And so it's still not very it's not a bridge yet But it will be a bridge actually it is a bridge now of course One other example is is a really really big Condip platform, this is the toll field which is located at around 300 meters of water In the northern part of the North Sea imagine here this structure Is more than 300 meters tall There there are no codes which we can use as engineers when we want to design and construct projects like this and We have to go in take into account all the uncertainties evaluate the risks and based on that We have to identify the right options for design the same goes for new and innovative activities or structures This is a concept for exploitation of oil and gas Reserves which has been developed say within the last 15 to 20 years where you take in the beginning they were taking old big tank ships and And then they were rebuilding them to use them as storage facilities You would locate this tank ship on one particular location Using an anchor system, and then you would connect to the tanks of the sheep some risers which are Basically pipelines going from the oil wells up to the tanks and of course there's There are no there are no codes There's there's no help to get when you want to to develop new concepts like this in that case you really have to To develop your own design basis now Here again, when we're dealing with structures like rockets We have the same problem. There are no rules no guidelines We have to develop everything ourselves Using the same sort of concept taking into account all the uncertainties making risk assessments. I Was involved in the design of this ring here for the IANA 5 rocket in the beginning of the 90s At that time the this is actually a picture of the RAN 4 rocket, but at this time This ring is for the IANA 5 and the idea at that time was that the RAN rocket should carry out the European Space Shuttle, which was called Hermes due to lack of money that was that was put on ice But I understand that this is now again ongoing So we the Europeans will have our own space shuttle shortly now We are also dealing the problems like management of hazards Natural hazards earthquake risk management When we're looking at problems like this We are looking at the decisions which are relevant to identify before for instance an earthquake Will take place during an earthquake takes place So typically there's some shaking going on and up until all the aftershocks have settled This is the process which we call during the earthquake And then in the phases after an earthquake has hit the decision problems are very different The information available is very different the risks In the three different situations are different and we have different options for reducing the risks So the management of these risks are A major concern in these three phases and again we use probability and statistics as a basis for evaluating the risks And our important aspect is in inspection and maintenance planning of engineered facilities Here we you see a bridge cable from a bridge Just outside of Buenos Aires The Sarade Brasso Largo bridges You see deterioration is something which can be really severe and on this bridge actually suddenly cable states cable stays started to fail and we had to Do what we can or what we could to safeguard these bridges here You see some corrosion on offshore pipelines here. You see fatigue crack In a in a welded steel joint on an offshore platform Decreation processes are very important degradation processes are indicated are also extremely uncertain And we have to take these uncertainties into account. We have to model them and we do that based on probability and statistics So I think I already said enough about the motivation for learning probability and statistics and I will jump into This illustration here. What are we really aiming for? We are we are aiming for decision making the basis for the decision making is an evaluation of the risk The risks are evaluated by considering the events which are associated with consequences We evaluate the probabilities that these events will take place now this Requires that we have probabilistic models And in order to develop the probabilistic models We we undertake what we call a model building process model estimation which again, utilizes data which are available And our focus here now Concerns the theory of probability This is of course one of the building stones. We need to understand Because I forgot that we are going from Aye, aye, aye. I'm sorry. I was I thought we were going from From eight to 845. Hey, I'm really sorry You were really sweet. You were keeping up with me without saying anything Thank you. Why don't we have a break? Can we agree to be back again in exactly 15 minutes? That means that we are back five 29 So what will happen now is that I'm going to introduce the very very basic theory of probability And I I can assure you that this presentation is very basic If you would compare to probability theory, which is being taught down at the institute of mathematics then You would see that there is a very big difference Between what engineers understand to be probability theory and what mathematicians understand to be probability theory but I think This level here is sufficient for us as engineers What is probability we all have some notion And we frequently use words like chance likelihood frequency and probability basically Meaning the same thing in our minds but Formerly there are significant differences Or can be we use words like this in in in connection with different activities Among which passing the exam here At the end of the Of the lecture is is also one of the activities and here I would say that the probability is very high If you work hard Now What are we concerned about? We are concerned about states of nature of nature Associated with consequences and we would in the end as engineers like to be able to evaluate the probability that for instance a bridge is failing Due to excessive traffic loads The probability that a water reservoir is being overloaded Overfilled by water Maybe in some context we would like to be able to evaluate the probability that a Power distribution system Electricity distribution system is falling out. Maybe due to a falling tree Or other interesting events like we have also seen here in switzerland the consequences can be extremely big Now just the event of a tree falling down on The electricity distribution system caused a fallout of electricity in larger parts of italy Or extended periods of time and of course the societal consequences of events Like that can be very big We need to we need to plan and we need to maintain our infrastructure Such as to minimize such risks Or an event can be that a project is simply being delayed In general, we are interested in quantifying the probability that such events take place within a given time frame So when we are dealing with probabilities, it's always a good idea to in your mind associate the probability with the time frame There are different interpretations of Probability, so the Frequentistic, I will show you three different Uh interpretations here the frequentistic interpretation that a probability Of an event a will take place Is defined as the limiting value of The outcome of experiments So imagine that you are conducting a very long series of experiments You're performing in total in experiments and during this process of experiments you observe that the event a takes place in a times And as the number of experiments goes towards infinity the probability that the event will take place is defined Is a limiting value of n a divided by the total number of experiments Now in this interpretation of probability The fundamental thing is to understand that the probability is understood as A quality of of nature, so it's something which is associated with the experiment And the probability Can only be observed it's something which can be observed and it's an inherent property of nature Without the experiments, we would not know what the probability would be So that's that's the conceptual interpretation of a frequentistic probability That basically also means that if we are throwing dices According to the frequentistic interpretation of probability We would not a forehand be able to say that the probability of getting one six Is is is one divided by six when we are throwing one dice one time We would we would have to do an experiment in order to find out. Okay So with this dice, what is the probability that we will get a six In the classical interpretation of probability and the classical interpretation was was developed by by People interested in in in playing games like playing card games The interpretation of probability Is is very different In that case you Evaluate Analytically simply by thinking the total number of outcomes of an activity or a game That would be in total And then you evaluate also analytically simply by thinking the number of ways the event a could take place And then you take the ratio between this number of of ways The event a could take place Divided by the total number of outcomes of this game And that is The classical interpretation of probability and then we're dealing with the Bayesian interpretation of probability Where the probability associated with an event a is simply the degree of belief By any which given individual so it could be any one of you the degree of belief you have that an event will take place These are the three different interpretations of probability Now in order to illustrate that let's look at some guy as flipping a coin The a frequentist guy He would be probably Some nicely looking person with with glasses or a woman with glasses and a white dress going into the laboratory and then under Very controlled climatic conditions He would or she would take a coin and flip it 1,000 times Independent of each other of course, but because this is a condition now During this process She or he would count up the number of times where we would have a hit and during During 1,000 experiments of this type She or he would count up 510 occurrences of hit Now according to the frequentistic definition, we would divide that by 1,000 1,000 is not quite infinity That's clear, but Also, we have to consider the costs of the experiment So that will come out as our evaluation and according to the experiment The probabilities 0.51 That and this is now a property of the coin Yeah, this is a probability flipping this coin That you will get a hit The classical researcher would Take the coin and sit down behind Her desk or his desk She would take out a pipe and fill it up with some tobacco and then for a very long time Consider what are the different possible outcomes of flipping a coin and after some hours Would be completely sure that there are only two outcomes Namely a hit or a tail And then again after some hours would come to the conclusion that there's only one hit actually on this coin And according to that we would get a probability which is one divided by two In popular speaking it would be 0.5 Now a Bayesian a Bayesian researcher would be some sort of Frustrated looking Man or woman who would be posed a question Yes, what is the probability that you would get a hit And and and the poor person would stand on one leg and say oh Basically, I don't know but then again. I also don't have any preferences. So I would say 50 50 And that would be a very subjectively statement in regard to this probability Now you will see in this lecture that we are actually in engineering Decision making we are taking the perspective of Of the of the Bayesian interpretation of probability, but And that's important of course we we want to utilize the Subjective knowledge Which every engineer and every person has But we also want to To be able to utilize Whatever analytical Information we can get in regard to probabilities of events and we want to be able to utilize Any information we can get from observations or experiments. So we are combining These concepts of interpretation of probability into what is called the Bayesian probability theory Now we need to Define a couple of specifics We are dealing with sample space and events and this is probably something you have already had in high school We're dealing with Assembled space the set of all possible outcomes of state of nature. For instance the concrete compression stress Test result is called the sample space Now sample spaces can be Can be of course very different For concrete compression strengths the outcome of test results is basically the Whole range from zero to infinity So this is a continuous sample space between zero and infinity A sample space can be continuous or discrete when we when we are throwing dices Then the sample space is discrete. We only can get one two three four five six With normal dices I would say Now typically we illustrate the sample space And also the events which can take place in sample space using what we call Venn diagrams And I'm also sure you have seen those This is this is the sample space. This is omega You can consider this to be the universe and then within the universe we have events Yeah now An event is formally speaking a subset of the sample space and we say that the the If the subset is empty Then the event is not possible. It's an impossible event If the subset contains all of the sample space Then the event is certain Now consider these two events e1 and e2 The in this case when we're dealing with such two events the subset of sample points belonging to the event e1 and or e2 We call the union of the two events E1 and e2 of course we can make unions of many more events But this is just illustrating the situation where we're dealing with only two events And we write the union between two events in this way So this is this union sign and if you look at it graphically in the Venn diagram You see the union Indicated by the red line around these two events. This is the union Now dealing again with two events We are all we also have the notion of an intersection and the subset of sample points belonging to the event e1 And the event e2 is what we call the intersection So we have these two events and then we have an area which is an overlapping area And that corresponds to the intersection With my nice little graphic here Okay, and the sign for the intersection is is basically the union sign turned around Then we have another notion to the event The event containing all sample points in omega not included in an event e is called the complement to e And we write it with the as an event with a bar on top. Then we know it's a complement To the event e So we have the event here e in in the sample space omega and everything outside all the green is a complement to the event e And it follows that the union The union of the event and it's it's it's complement is of course equal to the entire sample space The intersection of the event and its complement is empty That also follows Dealing with events there are some laws which apply We call those the commutative law the associative law and the distributive law and these laws They tell us how we can go about with operations on events Like the intersection and the union operations dealing with events And so they provide us the formalism on how to operate with events and they are very convenient to To know of course, you don't need to know these by heart After some training you will know most of them By heart I guess but in any case you can always look up how was it that we have to Operate with events mathematically correct Based on the on these laws The so-called de Morgan laws can be derived and again They look like this and and they are very they are very convenient to to keep in mind that they are existing sometimes It's it's very useful to be able to reformulate the events we are dealing with when we evaluating probabilities Maybe a reformulating reformulation according to the Morgan laws Provides a more convenient description of the event which we are really interested in In a way where it's easier to calculate the probability that the event will take place Okay Now the the theory of probability is in principle very Simple in the sense, you know from mathematics that we are dealing with all those axioms But in probability theory we only have three axioms So the basis is very simple They are due to Kolmogorov now the first axiom and You better learn this by heart Please Is that probabilities of events? They can only take numbers in the range from zero to one So a probability is a number between zero and one And if I ever catch one of you in suggesting that the probability of something should be 7.2 Then I cannot promise you what I will do But I could use the laser on you or something like that Now the probability of the entire sample space is of course equal to one So the event Comprising the whole sample space has the probability one The probability of a union of Events like we have here Is the third axiom is equal to the sum of the probabilities of the individual events as long as these events are mutually Exclusive So in case the events cannot take place at the same time The union Is simply summing up the probabilities of the individual events. This is a third axiom Now I want to introduce a concept of conditional probability We have been looking at a couple of important concepts here. First of all, we have been looking at events We have seen the sample space We have Introduced the concept of union of events The concept of intersections of events We have introduced also the rules The mathematical rules which we have to apply to when we are operating on events This is basically set theory The laws on how to do that we introduce the de Morgan laws And then we have introduced the axioms of probability and there were only three now The next step is to look at what we call conditional probabilities and this here is a picture. I took on my new seven mega pixel camera up in hüngerberg forest last weekend And it's amazing what you can find in nature and I am really a nature freak And I love to take pictures like that and then I sent them to my mother Because we have a very long standing and close relationship I wanted to tell my mother what I experienced in in the forest And so I also sent her this picture here And I was completely excited when I came back from the forest because it had been a nature experience and made your one Um, I have seen trees and animals and now you you may try to Imagine what I actually also wrote in my small email to my mother Um, what do you think that I wrote that I had seen in the forest? Come on Yes, that's right That Is exactly what you were supposed to say Because When I walked over behind the tree To look at the sweet little fox. I found this guy here and uh what I wanted to illustrate with this small example is that That when we are building up knowledge Then of course we are using what we see we use data Okay But data einstein said data is only information We combine data with what we already have in our heads Okay Um And in this case the data which is new here. We completely update our imagination So whatever we have in our head is utilizing this data and then building our world picture And and and sometimes this is a very good mechanism Uh, I wanted to illustrate actually by being a little unfair because this is I I did not see him and this is a picture I borrowed from my brother actually Probably it was a fox So we are combining information with what we have already in our heads and what we have in our heads is what we call prior information This is these are our prior models you can say we update our models Using whatever information we can collect and we do that completely automatically in our in our minds This completely automatic mechanism, which is ongoing Um And this mechanism we will of course utilize in the framework of probability theory So uh formally uh looking at the different steps in in such a knowledge updating Process first of all we need to have some sort of hypothesis about what does the world look like? And this hypothesis looking at at you as being human beings at least many of you Um is something we get through our when we grow up when we get education We develop a picture of the whole world Uh, and then we utilize whatever existing information we can we can collect In this process we combine it with data, which is runningly being provided to us And we we learn We learn how to develop knowledge coming back to the mathematical formalisms of uh conditional probabilities Conditional probabilities are of special interest Because they provide the basis for utilizing new information in decision making And the conditional probability of an event e1 given that the event e2 has occurred Can be written in this way and this is the way we write it So we write it the probability of e1 given or say conditional on the event e2 We evaluate that as the probability of the intersection of the two events e1 and e2 And of course we can interchange so we could also have written the intersection of the event e2 and e1 Divide it by the probability of the event upon which we are conditioning So this here can be considered as the probability of the New information we have the data And we see that this collapses If the probability of the observation is equal to zero In that case this updated probability does not exist Now we say that the events e1 and e2 are statistically independent if The updated Probability so the probability of e1 given e2 is simply equal to the probability of e1 In that case the two events e1 and e2 are statistically independent now From this relationship here the conditional probability as we had it before it follows that we can simply multiply over This probability of the observation Over on the left side and then we get this expression here That means that the probability of the intersection of two events is equal to the probability of the of the conditioning event multiplied by the probability of the event e1 given e2 and again remembering What was the definition of statistically? Independence namely that the probability of e1 given e2 is equal to the probability of e1 We insert that little result in this expression Then we in on this place here we have only the probability of e1 and in that way we see that If the two events e1 and e2 are statistically independent Then the probability of the intersection of the two events is equal to the product of the probabilities of the events Now in general This is what people always do if they're dealing with the probability of Intersections of events that they they tend to forget that the events could be statistically dependent and they tend To assume that when they evaluate probabilities now, you know better You know that this is only in the case that these two events are statistically independent We cannot assume that we need in general To evaluate this intersection probability in this way Now finally I will introduce a relatively important concept. We will illustrate this concept sorely Very sorely using examples and exercises This is called the the law of base or the base theorem consider this sample space omega we have here and now In principle, we subdivide the sample space into n mutually exclusive events e1 e2 up to em now n is a little difficult to handle graphically, so I decided to illustrate this n is equal to 8 So we have eight individual events Which we have subdivided our sample space omega into 8 now What I am particularly interested in is the probability that one of these events will take place Conditional on an event which has been observed So it's the probability of the event e3 given The prob the event a has taken place Now if we look at The event a first Then it's clear From this illustration here. I hope that the probability of a can be written as the sum of probabilities of the event a intersected with the individual events e in this Omega sample space So the first one is the probability of a this one intersected with the event e1 And we see that this is this is of course empty, but that's that's the first term It's not wrong to write it of course It's completely correct, but this contribution in this case is equal to zero now. We can also Add then the probability of the event a intersected with e2 So that's the red one intersected with e2 Again, this intersection is empty Yeah, but it's completely correct to write it The contribution is simply equal to zero then we can take the intersection of a and e3 Now we see that we get a contribution in the sum Again, it's completely correct to write it that way and we can do this for all the individual events e1 to en So this is merely a way of rewriting the event a probability So in the end Here in the next line The intersections of course we can rewrite also using the The conditional probability If you remember conditional probability here The conditional probability multiplied by the condition the probability of the conditioning event Is equal to the probability of the intersection of the events So this is what we are using to get to the second line So we are exchanging this here with the probability of the observation given The event The individual events multiplied by the individual events How could we do that? Is that absolutely clear? of course When we are writing the probability of this intersection here Then we can we can choose ourselves Which how we want to order these two events e1 and e2 So if I interchange e1 and e2 here, then I also have to interchange e1 and e2 here That means the probability of if we interchange now here, then we would get the probability of e1 multiplied by the Probability of e2 given e1. Okay, that's completely equivalent to what is standing here And this is this is how the second line emerges Okay, and now we can we can write this together using a some sign over the n individual events Now Let's keep base out of this equation just for a little while then As we have that this Relation holds as I just explained then we can write the probability of According to the to the conditional probability rule We can write that the probability of the event any of these I events conditional on the event a is Simply equal to and now now we go up here In this equation here This now and we can We can subdivide Over here with the probability of a down under here and then we get this expression here you see So the probability of any of the I events I going from one to n Within the sample space conditional on the observation can be written as the probability of the observation Given the event I of which we have interest Multiplied by the by the probability e I and then divided by this probability of a and here in the second location Or in this location. I've simply exchanged this probability of a with this sum we developed over here And this this formula is what is called the theorem of base and basis this guy here and he was A In the and he lived from 1702 to 1764 now base was a mathematician And you you can say that what he developed here Is is that mechanism for combining? Information so you you can imagine you start we are interested in the probability of a certain event given Information we have observed a the event a is the information we have observed Now we would like to be able to Calculate this probability to update our knowledge And the basis for that is a prior probability Of the event of which we have interest in namely the the information we had before we got the new information So that's the tree with the with the with the with the fox tail you can you can say Situation we are trying to model here This is a prior probability and this here this term here. We call the likelihood of the observation So given the true state of nature is Is the event e1 What is the probability of getting this observation? We are interested in this we call the likelihood and correspondingly the updated probability of the event we have interest in Is called the posterior base As a mathematician developed the mathematics for updating knowledge At the same time Emmanuel Kant From Germany a philosopher He developed in a philosophical way exactly the same sort of reasoning And that's actually quite interesting because these these two persons Emmanuel kent and and and base Thomas base they were living In more or less exactly the same period of time So one and I can recommend to you to read also Kant. It's interesting reading Uh kent from a purely philosophical way even using the same words as base prior information posterior information developed In philosophical terms the same sort of relationship on how we combine existing knowledge With information we observe and in this way Develop what we call posterior knowledge. That means added Added added knowledge Learning this is the process of learning now Looking at the interpretations of probability which are introduced in the very beginning Just be patient 30 seconds more We were dealing with a frequentistic Interpretation of probability We were dealing with classical interpretation of probability and we were dealing with what we called a patient interpretation of probability Which is a very subjective way of assessing probabilities Now what you can imagine is that using this framework of of base The we can imagine that prior probabilities They need not to be established using frequentistic interpretations For instance Or analytical evaluations of the classical interpretations If we don't if we don't have any analytical basis for assessing our probabilities And if we don't have any experiments results We can always use our experience and all the accumulated knowledge We have from other cases from other applications from other problems In a subjective way to assess the prior Probability and then we can combine this prior using the theorem of base with The likelihood we can evaluate from observations And then we can combine that means that we can combine data with Subjective probability estimates And then we get of course some sort of mixture between a subjectively assessed probability And a data-based assessed probability But you can imagine that if we start out with a prior and then we fill in The results of a lot of experiments Then the influence of the prior information, which was very subjective to start with Becomes Smaller and smaller as we get more and more data and in that way using the formalism of base We move from a purely subjective Assessed probability to a purely Frequent statistically assessed probability. So we are able to represent any mix of different types of information Yeah Okay I thank you For today, and I certainly look forward to see you in A little more than one week from now. I wish you luck with the exercises. Hang in there