 Hi, I'm Zor. Welcome to Unisor Education. Today I'm going to talk about probability and knowledge. This is a second introductory lecture in theory of probabilities for advanced math course for teenagers. The whole course is on Unisor.com and that's where this lecture is linked to. There are notes to this lecture. They are also on this website and they do recommend you to read the notes before, after the lecture. I think it's always very helpful. Okay, so this is a second lecture. The first one was about probability and frequency. That's one view to the probability. Another view is what I'm going to talk about today. It's related to certain knowledge about the experiment which is being conducted. So let me just have a comparison between random experiment and deterministic experiment. Now the random experiment is an experiment when you don't really know the result. It can be any one of certain number of results. Like for instance if you are flipping the coin then there are two different results. If you are throwing a dice there are six different results depending on what side is on top. When you are dealing a deck of cards among four bridge players you can have a lot of different distributions of the cards among the players. So these are all examples of random experiments. Now the experiment which is more deterministic is something when you definitely know that there is one and only one result and you can predict this result. Like for instance if you lift something from the floor and then just leave it to go down it will go down will drop on the floor. I mean this is something which you can definitely predict because the gravitation the laws of gravitation works and basically you can predict for instance the position of the planets which are circling around the Sun at any future point. So there are things which you can definitely predict and there are experiments which produce predetermined result. So these are deterministic processes. Now what's the difference? Well the difference is basically an amount of knowledge which you have about a particular process. Let's talk about flipping the coin for instance. Well yes you can say that there are actually two equally probable results but that's only if you do not take into account the beginning position of the coin in your hand, the beginning position of your hand, the force which you are exorging when you're flipping the coin, the gravitation force, the air resistance. I mean there are many many factors which actually affect the final position of the coin. If you knew all these factors completely you would be able to predict the position of the coin. So your experiment would not be actually random. So why random experiments are random? Well because we don't know all this stuff. We know only some major component like you are flipping the coin basically somehow but you don't know exactly how. You don't know all the circumstances around it. You don't know the conditions of the experiments etc. So the fact that the probability of both sides of the coin are half and half is actually a reflection of your lack of knowledge and that's what actually I meant talking about probability and knowledge. So the symmetry among different sides of the dice which are on top between different positions of the coin when you're flipping between different distributions of cards among players etc. The randomness is the result of your lack of knowledge about your miniscule pieces of information which are all contributing to this particular process. So you might say that let's say if you are throwing the dice then there is I'll try to put it graphically but obviously it's very very unscientific. So if this represents an effort which you are making when you're throwing the dice then there are hundreds of different very very small factors which you have absolutely no knowledge about which affect the final position of the dice and because you don't know this stuff that's why you are saying that you really cannot predict the result and results are basically symmetrical relative to whatever the side of the dice is on top. So that's basically my approach to the probability. Probability is a measurement of amount of knowledge which you don't really know if you wish something like this. As soon as you start contributing these factors into your equation of throwing the dice you might actually think that certain are more important certain are less important and after a while your information might be sufficient to predict maybe with a little bit better than half and half or 1.6 and 1.6 and 1.6 in case of a dice probability. For instance just for a chance you know that somebody when making a dice made a little mistake maybe intentional maybe unintentional and one of the sides is a little bit heavier than the other sides. Then this side would tend to be on the bottom and the opposite side would be of the dice tend to be on the top. So that brings the probability of that opposite side to a greater than 1.6 value. So this extra knowledge about how this dice was made actually gives you certain ability to view the probabilities differently. So probability being equal to all the different results of the experiment means that you don't know anything about how this experiment is conducted. And let me give you a little bit more practical example. Let's take the weather forecast. Now whatever the weather is right now if I don't know anything at all about meteorology about conditions somewhere else wind whatever. I can tell you that tomorrow there will be either a rainy day or a dry day with equal probability of 1.5. I mean that's the best thing which I can do. However give me some knowledge let's say I know that somewhere in Boston for instance there is a rain right now and the wind blowing towards New York from Boston to New York with certain speed which is basically about sufficient to bring that rain to New York tomorrow. I mean it might actually deviate from the course but still right now as we speak the course is towards New York. Then I can say that there is a probability that in New York tomorrow there will be a rain and the probability is greater than 1.5. So 1.5 to rain and 1.5 not to rain is just based on no knowledge at all. But as soon as I introduce some knowledge my probabilities are shifting. So that's why the probability the deviation actually of the probability of equal among different results is actually the manifestation of certain knowledge that you have about this process. About certain new conditions maybe which are before were not taken into consideration. And that actually brings to the concept of conditional probability which I'm not going to go into the details right now but basically I would like to mention that it's associated with the name of Thomas Bias 18th century English mathematician and the whole approach to probability and knowledge that probability is basically lack of knowledge and the more knowledge you have the more deviation from the equal probabilities you will have it's all relate it's all actually named after after Thomas Bias it's called biasing approach to probability. So it's completely different from frequency based as you see. So anyway I have this little plan I wanted to talk about so okay now under these circumstances we still have concepts of events and elementary events. However if in the frequency based approach to probability we assigned usually equal probabilities to all the elementary events and then after we basically construct from these elementary events any event which we are interested in we can just add up the probabilities and get the probability of the of the event itself. Like for instance when we were throwing the dice and we were interested in the event of the even number to be on top we just added the probability of two to be on the top which is one six and four to be on the top which is another one six and six to be on the top which is another one six so altogether it was three six or one half. Now in case you know something about the dice like for instance you know that the dice is loaded on one side let's say you have this dice and you know that the bottom is a little bit heavier then whatever is on top this number has a greater probability to be the outcome of the throwing the dice which means that my probabilities one two three four five and six would not be one six one six one six one six one six and one six as if you don't know anything about the dice but if you do know that something is heavier than another then let's say that the probability of number three is a little higher let's say it's two six and then these might be a little lower but the sum should be also equal to one because we're still kind of talking about relative number of one side to occur relative to the whole number of experiments which we are providing so if you know that the probability of this is two six and these guys are slightly less than that what should I say so I have four six divided by one two three four five so it's four twenty so it's one fifth so this is one third this is one fifths this is one fifths this is one fifths this is one fifths and this is one fifths am I right one two three four five now that's uh no let's put no one fourth I'm sorry one fourth four no no no two six four six divided by five four third is says two fifties that's two fifties I'm sorry two fifties so the probability of each other is two fifties we have five of them so it's 10 fifties which is two thirds and one third is for this yes so if my dice is loaded and that actually brings the probability of number three to be higher than one six with lowering the probabilities of all others then I can definitely say that well we still have elementary events obviously these are elementary events but they are no longer evenly distributed probabilities they have different probabilities I still can actually calculate what's the probability of the even number to be on top it's still some of this this and this but it's no longer one half it's one two three times two fifteen so it's uh two fifths so it's less than one half so again knowledge shifts the probabilities from being evenly distributed among elementary events to something which is a little bit not even depending on how much knowledge we have and if we have an absolute knowledge about everything then I can say that I can actually predict that the certain number will be actually on the top uh based on the force the air the the the gravitation etc etc momentum so if I know all that then the probability of some of them would be equal to zero and some other would be one that's the one which I can predict based on full complete knowledge about throwing of the dice so probabilities are shifted okay what else yes I was talking about weather forecast and it's actually very important you see before like I don't know 50 years ago our knowledge about the weather wasn't really very good I mean we didn't have all these computers models etc etc and the precision of the forecast we still wanted to forecast right so but the precision of the forecast wasn't that great as the science progressed as we have learned more and more details about how the weather is formed and we have all these powerful computers which can absorb all this information and process it etc so the precision of the forecast while still not perfect we are still not exact with our forecast but however we are really closing and closing to basically say that if if the forecast says that tomorrow is is rain well most people actually do bring umbrellas they trust this particular forecast and usually it does rain I mean yes there are some last time changes in the weather patterns etc however what's important is that the precision of our forecast is growing as our knowledge is growing which means that if 100 years ago I can say that the probability of the rain tomorrow will be one half now I can definitely say it with probability of 0.9 that it will or will not be raining tomorrow it's a much higher probability okay what else I didn't cover so symmetry I did cover the symmetry symmetry is related to basically lack of knowledge and as soon as we have some symmetry as soon as we have some knowledge the symmetry among the elementary events usually is distorted in some way or another one more interesting thing you see one thing is some kind of objective knowledge about like weather for instance another thing is about subjective belief you know sometimes the person can say you know what I believe that stock market will crash in the next three months period well to tell you the truth there is not much scientific knowledge behind this sentence yes there are some objective factors which the person might take into account however to say for certain that the market will crash in the three months period is not really the objective knowledge it's a belief belief based on something belief based on today's condition which tomorrow might change by the way beliefs maybe on something like I know some kind of desire to to persuade people to behave in certain ways so what I'm saying is that the knowledge and belief are two different things knowledge is objective science-based kind of a conclusion and belief is an opinion which might or might not be true and that's why you have to really differentiate before shifting your mental evaluation of the probabilities you should really differentiate between the scientific knowledge which can which can be brought by some kind of science and research etc and between beliefs of certain people who are trying to persuade you to do something in some way or another and that's by the way is very commonplace in in the stock market games lots of people express their opinions and sometimes they are not exactly honest people and sometimes they express an opinion just because they want to persuade people to act in certain way to buy or to sell some specific security and they pretend to be knowing something although that's not really the real knowledge it's more of belief or opinion or attempt to somehow change the behavior of other people okay and that actually brings me to another concept which I did mention very very briefly in the beginning the concept of conditional probability now conditional probability is probability of certain event if you know something about this particular event prior to making an experiment and as an example I'm not really going any further I will probably devote a separate lecture to conditional probability I just want you to understand what actually is all about let's consider the probability of the person to have lung cancer well it happens unfortunately people do get sick and some of them get lung cancer now there are insurance companies and insurance companies are covering your medical expenses so you're paying certain amount of premiums and in case you get sick they will pay your medical expenses right so let's consider an insurance company now if insurance company doesn't know anything about you well it basically thinks about this way well we have so many people a certain number of them historically had cancer during the whatever the period in their lifetime period so approximately we can calculate what's the statistics of this what's the frequency of having a lung cancer so let's say we have one in one thousand people average having lung cancer now the expenses to deal with this are such and such so what would be my if I'm an insurance company so how much money I would ask from the person as an insurance premium to cover expenses just in case he gets lung cancer well I should probably cover it in some way that from a thousand people I would give enough money to cover one that that's my average right maybe just a little bit more to have some profit out of it however is it fair contemporary science medical science tells us that if the person is a smoker then the chances to get lung cancer significantly higher not just a little significantly higher so insurance company would like to do it a little bit differently now it actually separates people into smokers and non smokers and they know that even if among everybody one in a thousand has a lung cancer among smokers it's only one in 50 but among others it's one in in a hundred thousand something like this I'm not sure about exact numbers but definitely the numbers are different so the person is asked actually before arranging this insurance policy the insurance company is asking are you a smoker and if he is a smoker they actually demand more money because the probability is higher now why is this happening from the mathematical stand view it's because the conditional probability of getting lung cancer based on the condition that the guy is a smoker is higher than the absolute probability unconditional probability of getting the lung cancer so if we're talking about all people without any knowledge about their smoking habits we're talking about unconditional probability to get the lung cancer and we can base it on frequency we have thousand people and one of them on average gets the lung cancer if however we add this condition the probabilities are changing and we're talking right now about conditional probability which is equal to let's say 150 for a smoker and one in hundred one over 100 000 for non smoker that's what actually the conditional probability is all about and these conditions are actually the knowledge so if we know about the person and we know about conditions of this person we are changing the probability probabilities of different results of our experiments well that's it I would suggest you to read the notes for these lectures again just to make sure that you understand all these concepts and that's it for today thank you very much and good luck