Probability Lecture 1: Events, probabilities & elementary combinatorics - 1st Year Student Lecture

[Music] good morning there's an 80% chance of rain on Friday at 9:00 a.m. says the Met Office in its weather forecast for Oxford an 80% chance of rain what does this mean well if you have lots of days like the coming Friday then you expect to get to it it to rain for 80% of those days that's an explanation that's useful to interpret the weather forecast and to behave accordingly get your umbrella out and don't get wet now mathematically speaking that's not a very good explanation though is it what does it mean lots of days what does it mean days like that one after all there's only one Friday this week about which this statement is being made so we need something more here particularly from a mathematical perspective there's an 80% chance of rain is weather random that's some philosophical questions here in the real world many things that we don't fully understand or fully observe may appear to be random and it may be useful to use probabilistic modeling there's an 80% chance of rain how can the Met Office make such a statement is there evidence is there data that they've collected yes there is they've got lots of weather stations all over the country they have devices on planes they have weather balloons that they launch and they get a lot of data and they analyze the data these are statistical questions right they can predict the future based on the past it's a rather sophisticated statistical application because there is a lot of data involved and also because predicting is harder than just describing and about the future in particular um well you have to have to use some modeling probabilistic modeling in order to um to make sense of the um situation that you want to predict now this particular situation about weather prediction is of course only one of many applications in the real world where probability um features and um it it's really across a whole range of different settings some very Elementary like tossing a coin rolling Dice and uh well shuffled handoff cards um gives gives rise to some some interesting probability questions others are much more sophisticated and weather is one of them but there's also financial markets or um genetics in in many ways s of how are all the species um related how how does the genetic Evolution happen in terms of DNA that gets passed on and that changes um both on a big scale um between species but also on a small scale from your parents to yourselves so there is um a lot of Randomness involved in that process and uh there is scope for probabilistic modeling in each of those areas it's some of it uh genuinely random others maybe not so genuinely random but we um we can still um perceive it as random and make useful um random models that that help us um take decisions ultimately now I'm a mathematician and you're studying probab um probability in the context of mathematics or computer science or both some of you with uh with philosophy um others with with Statistics um as well um but we are going to have to start at the beginning and really focus on the mathematical side of things we w't um to answer the first questions I asked how can we make um mathematically precise statements about probability and that is what uh what this course is about so we are going to introduce models um in a way that is a little less fluffy than this 80% chance of rain and uh we're going to to write things um a bit differently as a consequence so um so what uh what I want to do here is maybe call this motivation and then um say what I prefer is something like um well I mean this is uh I I put this in the context of Friday this week um but uh what we what we might do U more more generally is say let AJ be the event of rain at 900 a.m. on day J of this term right and then you can take 1 less than J less than n where n is probably 40 or 39 if you exclude today right so now I've uh um used some language that I'll get back to um call this an event um actually uh I want to call this an event in the mathematical sense that I will explain and then uh um the next um um point is that we want to assess probabilities now um um when data are involved is complicated but uh um at the outset we might just say um suppose the events AJ each have probability P right P being something in 01 for this to make sense right that's getting much closer to specifying um a model um but we need to say something more here and um um in order to fully specify a model um and that is some assumption about how these events are related to each other and the simplest way of doing this is to say independently now there are several questions particularly coming from where we started with Statistics with um yeah statistics mainly so there are several questions here um what's P clearly but estimating P from data is not what we do in probability that's left to the statistics course and then there are questions about assumptions notably um Independence but also is it the same probability each time even if you look at the weather forecast they tell you well tomorrow the probability of rain is less than 5% on Wednesday it's 20% on Thursday 10% and on Friday it's 80% so clearly the the the the Met Office um suggests models where where the P varies but we have have to start somewhere and this is the kind of setting that uh you will see most likely in in this course with events that are independent and have the same probability but of course that generalizes in some way easily but when as far as the independence is concerned not so easily I'm using words here that I will of course explain in due course um the word event um I will make precise um as the next thing I do pretty much um the word independent uh if you haven't seen it um is coming in uh much later or you can certainly use your intuitive meaning for now right once we have something like this we can answer questions such as what's the probability um that we have rain for K days out of those end days this term right and you can probably answer those with your School probability if you've learned probability at school um that'll be among uh among the kind of questions that you that you may be able to answer already but uh what we are doing here is putting some of the things that you learned at school on a more formal basis and so I would always encourage you um to think um how does this question relate to the course rather than just to what you know already because that way you pick up the formalism that may not be strictly necessary for the questions at the beginning of the course but where you will um need to be um more fluent as we as we go along now and there's a common mistake here in the word independent that it kind of has too many letters okay I think this is pend okay anyway right so I've written on the board it's my first lecture after a while I always have to remind myself to write in large letters and you can already see I started very large and got a little bit smaller how is it at the back of the room can you see everything that I've written okay thank you if ever I get even smaller and you can't please tell me because that's useful feedback useful interaction that we can have um in a lecture um so um so please don't hold back now before I continue with the with writing there um I've got a bit more advice here um my style of lecturing is going to be chalk and talk as it used to be called here's my chalk and I'm going to um write on the board and paraphrase what I'm writing but I'm also going to give some extra explanations and what I'd like you to do or recommend you to do is to respond by something you might call um ink and think um ink because it's a very good idea to take notes um and think of course because there's a little more than getting what's on the board onto your piece of paper um but the important process of getting it through your head and out at the other end is already a first way of processing and um you may well find that uh what you write with your own hand um you feel um more committed to um when uh when you're are trying to um to fully understand it and if you've written something that you're not quite sure about you can leave a trace and say say oh I want to get back to this right so um so I really hope you you can do that mathematics is a very active discipline you really have to do it yourself you can't rely on what you see in front of your eyes so um so make sure you get the most out of lectures by um by doing something such as taking notes if you can if you feel you're falling behind and you can't listen very well um there are other ways that you can try such as having printed lecture notes or I mean anything be done with a tablet or a style and a stylus um but you can annotate things usefully as well as long as you um stay active um that's the best way of um of making use of the lectures that we offer okay so let's get started with chapter one events and probabilities think I need more pens so the specific setting that we've had so far let's see what generality we can uh we can put this in so let's look at an experiment and that word is uh is uh sort of in inverted commas because not everything you might think of as an experiment but let's look at an experiment which has a set of outcomes and let's call this set Omega of outcomes and let's call the outcomes little Omega in capital Omega I've seen that your well for the mathematicians among you certainly the um introduction to University M lecturer um put a a video um online where where he writes out all the Greek letters and points out that some of them are more useful than others so certainly Omega the last letter in the Greek alphabet is something very prominent um in probability um and so um so I'm using those letters here so we call Omega the sample space okay the set of possible outcomes so what sort of experiments may you might you want to have in mind so here are two examples so um a if you're tossing a coin what are the possible outcomes well quite clearly two possible outcomes heads and tails is what they're usually called so let me write H and T second example if we are throwing Dice and let's make this a little a little bit more interesting let's throw two dice and let's make them distinguishable a red one and a blue one what is the set of possible outcomes of this experiment well in this case we are observing two numbers between 1 and six let's call them I and J put them into a pair like this and uh let's say um we want I and J elements of 1 2 3 4 5 6 okay so these are natural examples of sample spaces of of some simple experiments so once you have an experiment we can Define events certain things happening in the experiment and that may not only be observing a particular outcome it may be something a little bit more interesting and more generally we call a subset of Omega is called well any subset of Omega is called an event or Well for now um there may be some restrictions later but uh not for a long time in this course so again for example in the two settings where we have a sample space specified um we can look at the event that the coin comes up taals that is an event let's call it capital A which is the Singleton set that contains the outcome t or B in the setting of throwing two dice um we observe a total of nine that is the sum of the two um outcomes of the individual dice um is equal to nine and that corresponds to an event a as a subset of our sample space that has several members because you can achieve nine in four different ways if the first one one is a three second one has to be a six um you can have 4 five 54 or 63 okay so this is just uh simple definitions and terminology that we are going to use now we're talking about sets here so um it makes sense to just uh um use the language of sets um in our um in our course here um and recall some of the notation as well so first of all um if Omega um is an outcome and if this particular Omega is the outcome that we observe then we can talk about what it means to that the event a occurs so if Omega is the outcome we say that a occurs because if Omega is in a right quite naturally um so how can we use this terminology um well I've I've said these are these events are sets so we have set operations that we can interpret in terms of events so um the first one here is the complement of an event so the complement of a um and that is denoted by a with a superscript c generally here um in our mathematics course in Oxford some people may put bars on top or have some other notation um but this is uh what I recommend you use um now if a is an event a complement is an event as well because it's also a subset of Omega in fact it's complement within Omega that we're taking and what does it mean for a complement to occur it means that a does not occur okay now um we can have Union so a union a union B occurs if a or b occur or both and intersection a intersection B occurs if both A and B occur what more a set difference a minus B we can Define in terms of what we've already defined as a intersection with B complement um and uh we can say um this um event a minus B occurs if a occurs and B does not occur okay so here is set notation just what it it means in terms of events occurring or not in our um idea of representing experiments um and outcomes um in this in this way so one more notion here A and B as sets or indeed as events in our context um we call them disjoint if a intersection B is the empty set IE A and B well let's say if a and b cannot occur together okay so that's a bit of set notation we were on our quest of defining um a probability model so um I've made sense of what it means to say let AJ be the event of rain at 9:00 a.m. on any particular day I can use um suitable sets um to to represent that um but now we assign probabilities to these events and so um that is our next general task now um we assign a probability that we denote by P of a with a double bar so to each event a and that's kind of uh a little bit uh too too general for for a general definition and I think uh in any case I'm postponing the general definition of what is required here to um to Friday um for today I'd like to restrict to the simplest case case that is relevant to our two examples of throwing Dice and coins so the simplest case is where Omega is finite and all outcomes are equally likely now outcomes are the little omegas events are the collections of omegas subsets of capital Omega and so the consequence of the little omegas all being equally likely um is um that the probability of any event a is proportional to the number of elements so number of elements of a and that needs to be divided by the number of elements of Omega um for the simple reason that uh um one of the outcomes has to occur and so um so if you take a equal to Omega you want to get a one out of this right so um so that's our um simplest case of a um sample space that is finite and a probability assigned that is um making every outcome equally likely so in our examples A and B from over there with the coin TOs to begin with we have a sample space of size two we defined an event that we observe tals which um consisted of the single outcome tals and so we can then um conclude that with this definition and certainly in a natural way anyway the probability of observing Tales From A Coos is 1 ided by two2 in the second example where we have two dice and this uh um set of um outcomes that have a sum of uh of nine um we have a sample space Omega where every combination of um two numbers one up to six is possible that makes it 36 possible um combinations and uh the event that we've written out explicitly clearly has four elements and so in that setting the probability of our event did I call it a or b up there it's again a okay so um so that event um has uh um a probability of 4 ided by 36 and that is 1 9th okay in the second example I mean it's still a still a simple example but you can kind of see that um Cal Cal ating probabilities in the setting where all outcomes are equally likely involves counting and counting is a topic that uh um that merits a little more um um discussion here in in some generality so um so let us start with some Elementary combinatorics combinatorics is the mathematical discipline of counting in some sense and you can count lotss of different things a kind of a um useful discussion for us today is to look at what's called permutations or numbers of Arrangements of um of objects and let's start with Arrangements of distinguishable objects so they are all different and you want to bring them into into an order arrange them in a particular order so suppose we have n distinguishable objects for example the numbers one up to n but some of this uh um phrasing suggests that this is actually more General than that and so um so we don't need to be um any uh very specific here and any end distinguishable objects feed into the following um discussion we can ask how many ways are there to order them okay and a a way to order them is a permutation so for example you could just have them in the order one up to n or you could have them in the reverse order right but of course there are many other orders and the question that's being asked here how many are there okay and in order to work that out um you can kind of start at the beginning uh if you want to list them all in uh in an arbitrary order um we can first wonder how many possibilities are there um for what we put into the first position and there are clearly n options for the first object okay and once you've chosen the first object um there are n minus one remaining options for the second object okay and then after you've done this a few times um you can argue inductive and notice that if you've already chosen your first M object or say m minus one objects then uh there will be n minus n minus one remaining to choose from for the M object okay and uh um if you like this uh um kind of generality here um with the m um then that's uh that's great but um let me just write until there is only one option for the last object or let me say the N object right the word inductively is kind of a um a powerful word because um it uh it kind of um suggests that you can be um completely rigorous about this um if at this stage you prefer to say well let's just put dots in between um that will still be fine I think in in in this course certainly um whereas um the mathematicians among you in on the pure math side of the degree um will be told that maybe dots are to be handled with great care and we want to um kind of handle dots with some care here as well and um ultimately um anything that you write with dots and is a valid argument um actually um probably can be written out as a proper induction and hopefully without too much effort um but with effort that you don't want to spend every single time anyway the conclusion for us here um is in all we can uh um put all this together to the to obtain the number of choices that we have here and there's one more um thought we need to have which is what do we do to n n minus one down to one um if you start at the top after the N options for each one of those n options you have n minus one options so Al together for the first two positions you have n * n minus one options and the same reasoning applies inductively um to say you need to multiply all of these numbers so um together there are n into nus 1 1 um into 2 * 1 um which is n factorial permutations right and let me write the word inductively here again because writing these dots um there is something that you can make more precise so um as an example there are six factorial ways to order the letters of a six-letter word and let me use the word gallowa here which is a famous mathematician which among his many distinctions um has the distinction um that that he has six different letters in his name and uh of these six letters there are three vowels and three consonants now that of course is uh is not uh particularly specific to this name um but leading on to my question that I'm asking in this uh in this setting um um if we randomly reorder the letters what is the probability that the vowels are all before all the constant consonants okay I've been a little sloppy here um so here what I really should say is um uniformly at random in the sense of uh of what I've defined up there all outcomes being equally likely so all rearrangements here um being equally likely then here's the answer how many ways are there of achieving um this event of having the three vowels before the three consonants um there are um 3 factorial time 3 factorial ways of first ordering um the three vowels and then ordering the three consonants um and that gives gives us 36 arrangements and what I've just said is of three vowels and then three consonants okay so I've done some some ad hoc calculations here so let me maybe hook onto the notation a little bit better here so um so if Omega is the set of all arrangements and a subset of Omega the set of arrangements with all vowels before all consonants then what I've calculated is that the probability of the event a which by definition is the number of Arrangements that have all vowels before all consonants divided by the size of our sample space well that ratio now is 3 factorial * 3 factorial / 6 factorial and if you work it out 36 / 720 is 12th okay so you can try the same with some other words um and of course um if the number of vowels is different um then you get a different answer okay so this was the simplest uh um setting in which we can ask how many arrangements are there to order objects um when all the objects are indistinguishable so let me now um move on to the more General uh task of finding out um how many Arrangements there are if not all objects are indistinguishable so Arrangements when not all objects are indistinguishable okay let's start with an example so how many different arrangements are there when I give you six letters is not all different so different Arrangements of a a a b c d so the letter a is repeated we have three letters a and the other three are are distinguishable so does this relate back in any way to the previous problem that we've solved we can kind of uh make some slight modification and say well I've got a a and a but if I had um three distinguishable varieties of a then I'd be back to um the problem that I've already solved so if we had A1 A2 A3 b c d then there would be six factorial arrangements or orderings okay so what's different now well we can kind of look at those so let's look at some examples for example um we can look at B A2 D A1 A3 C or we could um say yes but that's not the only way in which we can have um b a d a a c we can kind of um have different indices to the A's and another example is B A1 D A2 A3 C right so they correspond to the same order ordering of a a a b c d and so the question really is how many of those are there for each one um that we that we write down so um so what you what we are asking is again something that we've already answered in some generality because we are asking if we want to count how many words are there with A1 A2 A3 in any order but b d and c in the places where they are well then it is three factorial ways of permuting the A's among themselves okay so each one is one of three factorial different um which differ only in the order of A1 A2 A3 so our original question was how many different arrangements are there so um each different Arrangement appears three factorial times if we count all arrangements with distinguishable A's so um the um six factorial um orderings fall into groups of size three factorial and we are interested yeah let's say which are indistinguishable not with A1 A2 A3 obviously but uh if A1 and A2 and A3 are all identified with the letter A we started with then uh each group has has size 3 factorial and we want to count the number of groups okay and this is 6 factorial ided 3 factorial okay well that's an example an example of a number of objects that are not all distinguishable so what is the generality in which we can make uh make similar statements so here is the generalization so if I'm interested in the number of Arrangements of the N objects which I denote by little X1 and I'll have uh several little X1 um so um let me write it in this way I have M1 uh copies of little X1 and then I have little X2 for a few times and let's say there are M2 of those and then I have some more and finally the last um object is XM and no not M mxk and I have MK of those okay if this is to give me a total of n then n is M1 + M2 plus Etc Plus Mk so the number of Arrangements of the N objects where um we have M1 x1's M2 x2s Etc MK and K's xks um well let me write it is n factorial divided by M1 factorial M2 factorial time Etc time MK factorial now does this need proving I suppose it does the kind of proof that we've given to work out that there are six factorial um orderings and then there are six factorial divided by three factorial orderings in the um indistinguishable case and in the case where we had three the same um that same kind of reconing can be extended because now it's all about saying um let's make the X1 up the x1's distinguishable let's make the x2s distinguish we have all extinguishable objects then we have n factorial that's our numerator and now within each of those um um Arrangements you can permute the x1s you can permute the x2s and you can permute the xks in M1 factorial respectively M2 factorial respectively MK factorial ways and so um a group therefore consists of M1 factorial time M2 factorial up to MK factorial um different Arrangements of distinguishable objects right so same reasoning therefore the number of groups is this ratio there is one case that's particularly important and that is the case k equal to 2 and if you haven't seen this General case um I think you have seen case k equal to 2 so the number of orderings of say we have v's and X's or let's have such X's here and let's suppose we have M of those and N minus M of these consequently um is n factorial divided M factorial into nus M factorial just by applying the formula over there in the special case but you recognize this as a binomial coefficient of n choose M and also see in this symmetric uh expression here this is the same as n choose n minus M right so this binomial coefficient is a special case of the uh calculation that we've done over there the the outcome of the calculation so this is um the binomial coefficient which you may have denoted differently so this is n cm in what many schools use as notation but we very much like um this way of writing the bin Al coefficient not the the other way okay so um that binomial coefficient is uh is something that you're familiar with and it is useful in many ways and not just in order to find Arrangements of objects but also to choose M of the given objects from the total of N and you can do that with ticks and crosses for example to work out how many ways are there to form a football team of size M from a squad of size n just by uh identifying uh a football team as all the players um where you assign a tick um whereas the um other players are assigned an X and so counting how many ways there are to um select a team is exactly the same as the number of ways to order the ticks and Crosses but I haven't got time to write that down so I leave you with that um to to maybe think about more if you haven't um put this together at this stage okay so this leaves us to say well there is an 80% chance of rain on Friday well I'll see you on Friday at 9: anyway [Applause]

Channel: Oxford Mathematics

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Keywords: maths, math, probabilty, Oxford student lecture

Id: 53nZY5udKHQ

Length: 51min 10sec (3070 seconds)

Published: Sun Jan 07 2024