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Probability (1)
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Title  Probability (1) 
Title of Series  Introduction to Probability and Statistics 131A 
Part Number  1 
Number of Parts  16 
Author 
Cranston, Michael C.

License 
CC Attribution  ShareAlike 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
DOI  10.5446/12880 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Mathematics 
Series
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00:05
Klaus I had but assault do the interruptions Ghailani was Christ and my office is enrolled palm 5
00:36
40 B But Look for But But Office hours at these times does anybody cannot make a Idaho sides point person to take on Friday also then Nam Wednesday are Treacherous But Still him again It's a For your fulltime and review bill because of you try right For There's a lot of information on the border be things post on the Web page But But it would be a stiff but if he it But what was all So br Assignments will be to Friday's except last week to Wednesday Because the final exam will be the last Friday of class for greeting Plant Rodrigo we determined by Following paper sector were lying on the result of court which is very real Temper set will be I found quizzes At least 80 per cent from 35 per cent will be based on the midterm and 45 per cent on the files at all At the top of Victor Louie Friday July 12 and will be in the discussion section and the final will be in this better debate in this room By the way He full but now All will work But But the only of hope And you can't see that that him better K The syllabus is on the Web page that you It was
05:29
Really OK I think that examines the very 1st That he altered so bright start thanks very much From That I had 31st right OK um They also Video material covers all lying John whether pictures is still the telephone workers cover 1 week 3 or 5 essentially Each week we go through rumored 2 weeks of material from the standard or with sprinklers so the pace the course will be for quick maybe like that that would mean that it would be easy to fall behind you don't keep up every day with the material solar Anchorage to study hard to all the assignments Homework era assigned and even do homers that's not assigned and Mora of said you through doing homework problems the barriers quarrel and so you will manage to do a lot of the problems are from the chapters of the quite well armed the them from time to time I'll put up other things on the media outlet page maybe some biographical remarks about some of interesting statisticians what 1 of these statisticians maybe next quarter of that Will study rhetorical study work for obvious brewery And maybe no abuses is actually bigger Agra business and He published somebody from has published some material that was considered confidential and companies like it so after that They forbid are researchers from publishing and the sky had some interesting material that is thought to be published published under an assumed name and the name user student and now they're there's something called a student statistic even get that this quarter I think that's about it for a few drops in any question on these things procedural gaps organ the recitation Wednesday's so on starting this week end Al bout the the usual civilized policy of dropping the lowest score on a quiz Quizzes and score homework are what are they of us all as discussions Now I'm sorry don't laugh Other question fights class goes for 2 hours from typically also called for a parade around 10 minutes through and will resume tender outright finished by 10 45 sometimes it's archery stop talking once I start ties be reasonable also were rules please don't leave during the cost us you have Some of us in Parliament that you have to get to know texting during class I will call you tax I see it also know surfing the Internet No computers out I'd like to have your full attention class will be so it anymore Procedural questions yes OK let's see you go to work
10:45
I don't have the address on the top my head but here's I find it go to the mat Web page so that's Mass start UCI EU and The top there's a bar and no 1 of the things areas under graduate click on undergraduate and then Over on the left and think as courses courses and I think that should take you something is that course home pages The cat forceful later Zenobio of all classes that being offered this summer when you think of And media firstorder will be up later today are soul and the topic of this course is probably statistics
11:56
Trouble is a very useful field comes into modeling all sorts of phenomena in nature science and were studied were starter study of mathematical on aspects of that subject End tough for All old polls Judith Our probably starts with some legal the sample space that describes all the outcomes comes in an experiment so the most basic experiment our problems have in mind is quite tossing Back but also could both back EU food toss a coin once there too Onslaught comes a year it said Urine tailed shows up and so here's sample space of Now I'm sorry column consist of 2 possibilities Answer tales more that by 8 Shorty The That would be what they are sample space would be if you toss a coin wants what you toss a coin twice But I don't what to do But bad Over the past bottoms we now have had bullet times that had followed by tales are Sales followed by Lord Both jails And we keep going like this multiple times all heads and Myers followed by a finely Octavio Maybe the penultimate toss was the tale but although their heads And finally at the end of this As we have all tales of sample space can be quite large depending on what vary novice could also armed there may experiments that have 2 oil drums All All But But Who will cut should Seoul sometimes we might say 88 is sat with 2 elements And to Billy R. stands success Now stands for failure
17:11
So this might be when you're testing medicine you you have the medicine to a subject or something and outcome will be asked if the patient recovers from whatever ails fur and failure of the medicine didn't work or you might send an email These success if it reaches its intended recipient failure doesn't so this is of great common sample space and you could have repeated trials Mr. B S S S & M and half have all or you might have multiple trials Sweden not testing medicines you would not put much of Europe went But much weight on it Paramedics were you test of the 1 subject but you might think medicines prefect of the test thousand people work 900 90 times or just wants well who knows whether what that means That union larger samples of agreed Come to sap spaces like this Call Her ample Bob roller died You might enter that isn't done this But The sample space If you are you viral tour And you wanna keep track of multiple walk With I got severe sapped space with 36 Are you might roles landfall . 5 of them and then you'd have a larger sample space where you keep track of the year come on each Ross's means that are the 1st solely 1 several never won 1st won several Sir Also is Or coach or all but up to now Smoke detectors work by alphaparticles nasal capacitor when their small small bits ionized No sparking jump across faster making a circuit Perhaps You might count the or else particles that is emitted from your smoke detectors at some number over the sample space beers beers Various account number of apart from Hong Leong particles might have animated possibly none Maybe 1 maybe 2 maybe 3 to restart anywhere I stop somewhere well maybe admitted more so I think we should have the whole of natural numbers Each with forced back This could also be used to model say the number of customers that come into 711 between 12 and 1 in 9 Taipei on a Saturday morning a community Number of customers command or the number of email messages and received in a week
22:10
Some like that there were times lightning strikes in the field That's a 5 acres big in the middle of Kansas or the number of tornadoes in a month Nebraska anything anything that's counting Things like guess harmony bosses come by in an hour Back when I 1st Sigh All Sometimes you might have to wait for a a boss How long might yet wafer of us well I don't think a negative number of the appropriate way minus 1 hour or that would been so here's the samples they should be something like That And probably some you think I should include infinity there You were ready for a long time for us to wait for What time because you're here now I'm still waiting for a bus also army you drive probably wanted revenge for like 734 car back at a red light
23:55
And tho Seems like dipped waiting for them on time then but what we have to wait for what well the 1st car has to go person 1st cars on the phone doesn't see the current 2nd citing car has to go and that person's texting In the 3rd car drivers turned because his little Sundance crying in the back of it so you have To add up a bunch of waiting times and I'm sure you're aware that so that's another waiting time will study the physical properties of these later but again it would be Or even in this instrument here you might count the time until the next alphaparticles emitted notably sewer there may be other things 1 Come on Potrero not overhead is so a sample space here would look like what The On overhand has 5 cards and I believe and dumber you might have something like a beast of spades seeking his spade queen spades Jack spades and diamonds resided state flush and etc. pretty calendar arraigned priest sensible Public sample space for They enter goes on not there many possibilities for it so arms The race things with sample space Osama their sets it apart The key for which will be Subsets of sample spaces are Now you have to be a little bit careful there's some technical conditions that we have to apply here when the sample space is something like half line or maybe the whole realign but I will get into those technical details for Leave other sample space is a very discreet like this there's there's no technical problem about what kind of things can be a subset in speaks space that is something that can be put into want on correspondence with this will be called an event that saw some examples From from 1 baby out SAA aid that the events that you had 3 heads on Tossaint . 7 6 times I would describe event will be a subset sample space rewrite string of it isn't she's length sects and this 1 would have a following elements Just look at how do Maybe By Tom alas upon that OK So does occur question how male answers that You don't have a mail comes right Through the 6th I think tongue army outcomes are in the whole sample space We'll do this we were a startling counting a moment that can you so Set of all possible outcomes With be The 6 where but having half had set partner sex 6 now at Beth 16 Audi 60 Gap 6 slots will do that we're aware of this starting today where to start counting only things Satellite like this so there 6 slots were put in an age a these represent about comes on the 1st 2nd 3rd 4th 5th and 6th talks the way you do this is a victory Places to put and the remaining at wouldn't so I lose can't humbly waiting picked 3 slots would endanger the rest is automatic no choice after that he and remaining so says in the back there is there's something called 6 2 3 It's a number of ways to Who's of 3 elements subset of insects all my years are 6 elements at 6 slots and with free from there and in those 3 will put in 8 so some age On possible Tyson the remaining ones have to get I hate Well if you're both amid a bill laughter so more but they answer period Can be written 6 2 3 and it's a sixpack Royal over 3 factorial 3 that's how many choices Event Outcomes are not so Adoree donor centers right now we're doing this today grants
32:13
Um maybe a Eagle a straight flush with big and that straight flush means do did consecutive cards On the same suit summer the rule August texting You don't you get all the cards in the same suit means all part Or they World diamonds rock clubs and consecutive quarters might be 2 3 4 5 2 3 4 5 6 of hearts so this is a rather big as teams are used the big regularly a ace of spades king of spades Green a spade jack spades Tennessee states and then maybe playing a spade screens dates back dates and spade 9 and etc. and I guess maybe the area lowest 1 would be something like I guess the eastern BAR folk art and Booker 0 1 2 3 4 This would be the lowest rate past every other state fostered Swann Sorry over What comes in in 20 orchid How would you are Cunningham where 4 suits right It's always had to start somebody there are no particular thing in a mood to what multiply that number by 4 nominee a straight passes are their focus and states 13 well contact how what counting the lowest card Only possible pop up comic possibilities are there for lowest are strict Well we get tan is 1 possible possibility 9 is 8 All the way down to 8 at Eleven 10 comedy numbers are there from 1 to 10 10 so their tent in a particular their forces that means fought sold YYA on Full House ordered this event this means 3 0 1 numerical value to of another summit 3 aces in the 2 kings courageously and then maybe 3 rules 2 aces treating not you're playing poker which is out of false or a strict twice at because it wins right a wide is a winner It said This event has laments that this this is less likely to happen why would this be less likely to have happen lowered the with little likelihood depend on harmony elements are in the event in this case because which hand is more likely this 1 or this on saying What about this 1 or this 1 The same so attempts poker 85 partner and has the same probability as any other cars so if you have an event that 2 events like straight policy which 1 wins Which work Everyone has fewer elements straight flush wins over a house because you're straight losses will come this Later OK in waiting times so that might be something like your weight was less in 1 hour Adobe interval from 0 1 And finally are on to to the truth The full force of the sport at all But a Was with all the force which teach if They might be that Successes were observed perform some experiments and times and you see that Kate times you have a success Other times red failure They could be any number years from 0 might have no successor story every trial might result and success For example the experiment might be tossing a coin And then they would be the number of hats You might consider that a success if you're playing day worry when DOLLAR from an opponent every time ahead comes up and use a dollars for every time what's probably that had comes up with tossing a point at half mast time depends on who's tossing point I know someone who can toss a coin have every time and that's just an ordinary point he was a magician and became a mathematician works probability noise in the National Academy of Sciences Year's many skills 1 of them is being able toss point have a cockpits every time you can also do this he can take a deck of cards Just purchased opened that you take us off public 4 times a shows you it In the natural order same order came from Mr. what you get into It now maybe we should do a little bit of remind Surinder aboard
40:40
Come combining I sets since word Kmart events will do things like that unions intersections complex on is remind you briefly but those center of the union of 2 events would be the set of things the sample space there are quieter and Maori and we don't have to be both Victoria Way Looks like this And that the union would be this entire Region enclosed and Thanks Otis varying inside years as set a hearing insiders said beat a union leaders and plant Yeah section That is what we get if we change the word or 2 and Access to be in both sets to be near section of my victory and this being a and this being and near section is just Where they overlap and a minus V would be ever things sample space that acts as if they were and X is not in
43:22
It is picture of Sybase years 8 Here's being what is it What about things RNA but not in He is a wretched rich but it also do be minus a never be earning over here so Supply Co at the moon during What This is Chatelier's and another 1 Farm couple Marguez But a comment that is a set of things that are today So all picture the sample space is being box and an 80 years This thing here Not a R a compliment volunteering Not enclosed say there's a doddering stuff I defined as set to be a minus being union be Meyer said Ms. metric difference He put it What is So far I draw a bead of renewing before What would this Look back and say well what is a spotless but as Red region here Or the other said look like you're whatsis when he might say I stated in this part over here Now I take to order or Yeah every but That's they saw as would be better End up with is a symbol for the empty set But for But Which has of course no But now probability what we do is assigned values to elects dollars value probably it's number between 0 and 1 Each time But what it Now But for for But he also bore is as old as port or or both All At all As if it as if all of that shit The cost of all but over all of the old or but but but what we do with all of the was as strong Mitch later strike means As but Post They have no overlap AI intersect AJ which we defined Is empty and no 9 overlapping sets a probably the union is a sum the probabilities That's where a probability and I should also say 1 more condition and the probability of the whole sample space is white Back online If you like that 0 0 I'm just not as calligraphy and 3rd just make it would like serious capital letter he I could even some people even make it look more that's crazy OK I wanted make it look like it's important I didn't do that would have tipped over Terror base Hey so on busier empty said says that Yes it said Right Despite definition right Any subset of the sample spaces at events so cell number assigned to the empty set what number would that be wife correct Is probably that nothing happened but
52:27
Maybe doesn't convince you that problem that is 0 because must that some evenings in your life and nothing happened well let's do their sets Of Menton are samples basics like that and Saigon event today it will be the stuff here what what a Koppelman look like And now fared so sigh Drake What's the union of AAA copilot on parenting right by put those 2 yellow mega watts intersection of these 2 for you can't have something as both in May and not in a simultaneously so I take a a an 18 2 ABA comment and apply this swears it saying says they are disjoint Best Right that's workers as saying but would enable it to Tim what's a over here But the number assigned a union accompanied by 1 because this America so the probability of me a couple of years 1 Myers probability of a OK this is really really trivial yet it is very useful because Tropic or CBS review probabilities what's a problem that's what's problem that sometimes it's easy to computer probably sometimes not might not be easy to the probability of a massive computer probably of something that's large larger involves a lot of counting if even a small man Green is probably might be easy so if any is being what they say about a compliment Be small right so this is a way of computing probative is probably a couple is really simple Computer beside might be simple but this might be hard to beat directory 1st But indirectly through this case even those is terminal They have to be very useful can convert a difficult competition and 2 easy OK go So let's take a 10 minute break you're from back it off minutes gave us a resume due to more things Sunday probability and they go to Some Kartik was That We are Expos Ip 2 events Thursday as there's a mail What we say but However These probabilities The say this is required IT prove that AT reasonable we're probably should have this property but does it follow from a definition as my additions are really pain sometimes you prove even the most obvious things the reason is because sometimes the obvious things are right Fox River but to be careful Um what Properties we have trouble is well aware of this 1 and this 1 which would be good useful in turn that was under house maybe not maybe this this 1 says only work when near a union of destructive acts of somehow we wanna get unions at start a new and but there are other properties like maybe this 1 might be useful so what would be Destructive and being Well maybe how about this A and B minus a job picture of folly of a year already The staff would be minus 8 That part right are disjoint because aiding in here can be an India has been a solar development can simultaneously in both the union of bills By foes together where we get Now we have the union destroyed events we didn't apply Less to 2 events which to this 1 and this 1 applying that property over there this is Probably a answer probability of the United States and Eastshore equal over here we probably be so nor the middle part near where we get this probability because probability is a swan Alcoa which oversees quality of care This is a miracle a 0 right Can't be negative so that number has Small or Assad to work with Brooks I called back down Well bed last more The value of all you would know is what already is equal to that but Why couldn't probably be larger Although could be that as the U.S. go on so long but We don't know this probability is 1 But you might your idea would work if you divide giving by probably be and then Oregon and consider rolling subsets of Onassis you could do it that way
1:01:53
I would work at think this is pretty direct OK so I think that's If for basic things about probabilities are all good I was counting She met or or or 4 Or she for all of them it all The so it since several sample spaces worry might wanna say that they probably every outcome the sample space is the same as every other opt for example of a simple song was the sample space of heads and tails bad if you have a fair coin probably heads is a half probably tales that if you order the more complicated sample space were you may be costs coin a hundred times Demand Any sequence 0 my hundreds of pages and he would I have the same problem as any other sequence But fight for maybe confine myself to a stop But But for which the opera was more likely to pass a fair point 7 types They're the same same would would have the same probably nursing like that for a if I drew any other sequence of . 7 of 8 isn't probability that sequence of these differ doing successes and failures and each 1 has public half of happening and In drawing cards with Jack You got 5 cards a deck of 52 Any handover composer 5 cards is as likely as any other composed fight so the B 2 examples of samples of sample spaces where these probabilities you want side events are of the same and if you have bad situation But as good Opel full Gained from now on whenever I wanna write such that America is a British revision ST Because resist much and I don't like to write so much that is also comes ever same probability and have event in such a sample space than the probability eh Just BV A number of elements of a over the number of Robertson a sample space Because what's a probability chopped on Sept or the probably reach out from about let's enumerate the outcomes of suppose we have wherein said sample spaces pass by property Romania be infinite As for worse where what be bags there are compose over just a single prompts If you're Are these events disjoint for different eye AII intersect at age is empty where lot their union For whatever SAN or But say this number appeared and that Could be infinitely fast but I'm not ruling out So these are mutually disjoint 38 years Omega so what's the probability of union there to things we can say but the problem Union 1 is probably mega which is on And then there's these are mutually disjoint as B the sum of all the individual outcomes probably there may I happened right away from reaI promise you this but I want And how worthy different some ends here different They're all the same the same value Harmony of the Mars And Sosebee times of probability of may be the 1st one so what's probability of a major 1 That Solis using the extreme ends of this string equality probably Romania 1 probably this Individual outcome would be won over And the number of elements and has a number of elements and is a true but I've too because they all have the same problem of all comes at the same probably we know 2 things are the samples space they can't be infinite why not influenza was 1 over 0 more than that The Probably will make you would be 0 we can that fall comes and have the same problem we must be talking about a fine example space and then the probably beats outcome is 1 over the number of elements in the
1:11:22
And now why would this be I would this be once this is a number of elements Semarang With computers probably what odd thing we know so far is probably the union of destroyed events as some of the problems analysts next knowledge right All these I'll comes in Omega I'm sorry in a food with some areas may be from a 0 1 through maybe they are different from the different elements of the sample space and I rate rewrite this as union I equal 1 decay of the sets that contained the individual comes cracked Know what are these 4 different On differ Oh maybe Eisner from major to censor will be distract right here's expression of a hazardous joint Union of events and suddenly say but probably a it's a some of the probabilities of individual outcomes in that thing And now these all have the same value would survive released its 1 early The size of a major right As for 1 over the number of Romania we add up this that many times we get Cheney over the number of elements make awards K is a number of elements of It at all So the computer probabilities in sample spaces where all the outcomes are equally likely be probably went on to count the Romans and event Member elements in sample space take the that fire will get some techniques for counting a question on this As allow us to do they probably lost or to pair for 3 times over and then you'll understand why hands Our ordered away are objects 0 All the accounting error that will do Comes from 1 simple principle But it open he or she Walter occasion principal at with it But Amid changes to Amhara An a day notes without much of it on E for all but 4 of which a Southard Wasted to perform 1 actually be and and waste former 2nd action and there are times in waste of former both Show that At 1st I Gaza coin Darryl died so terror was tossing quite too is died Omri possible outcomes are there and the 1st experiment to Camarena Seguin 6 salt Amalia they're trough will on or case sewer You've probably been situations were Had to select 3 people from a class may be 1 hall monitor want to collect the homework and never want to be the bathroom on back from right there 3 distinct Jobs have be done so on their 3 individuals for this and I don't know how many people are here but I would guess there are 47 people here I come so are his job was up 2 . 3 and so I have some water here like its there 1 or off the 1st thing I do is pick someone for Hall Monday when you going kids are going up the court order casts Hollywood possible choice is going out for a month with 47 Now Life The 1 I picked the same person again so I had to do former 2nd task comedy waste paper from the 2nd test 46 and her desk Eddie 45 K So that number is well tumult on the other hand orders cases is never Norway's slack 3 people from a class of 47 and form an ordered sample Kansas He took but He laughter look or she off course No orders importer here because I was different tasks to each person selected What like it same through prior to tour more passed through same person The resort of the same person they're all 3 roles or maybe 1 person could through their jobs and There was a fair job comedy ways of everyday do that Sigh of 47 choices for the 1st one but now I can choose person again so I still have 47 the 2nd and I would stop for for the 3rd as a number ways deformed or select
1:20:52
Use words select and ordered sample with replaced from a group of 4 itself for set We don't want them all day That's what ordering whatever weightier drop ordering from where fighters want a committee of size 5 from from the past Do 4 4 4 4 4 4 0 over with it Oversaw as a study whether without replacement I did not Without 95 different people did so on what if we did this kind of thing here these 5 and I just do well a 47 waste pick the first one 46 of the 2nd 45 The 3rd 44 4 and 43 raised us 1 is that going to be this summer it's over countered by how much well 5 worthy individuals might affect you you do but might fit you you or you you you you you So Harvey ways of Iowa over countered that the same set of people I met pick her 1st 2nd 3rd 4th 1st 2nd 3rd 4th comedy ways of we overcome Yet the number well that's exactly right and in words that the number of ways to order Objects calmly ways can you range 5 objects were 1st objects not about stuff like that on calmly the range by people to be the 1st person 2nd 3rd 4th or you could be 1st Mets can't wait to divide by the number ways of rearranging comedy ways are the arrange 5 distinguishable Are up as a Scot Leazer copra mutation All so wrapped then yes that he for I and all or But Comedy permutations are of the 1st and numbers and a voice can arranged covers Well is uses a lot of patience principle may be an extension of a from Truda and things already with choices are therefore The 1st thing And and every time I make a choice of something for the 1st position Army forces remain for 2nd position and unless 1 and and times and my too Our position and might 3 for the 4th etc. and their 2 choices for the next last position how meters us well that's the last player and And by multiply all these numbers together there's a nice thing for that number What is it And you would oversee factory orders and with a lot of emphasis because that's end A and factory fights Let's go back to this problem we will recounted here for every tours of the same people we can't do that Calmly times number ways we can order 5 people army wasting your 5 people 5 factorial social divide this number by 5 fact for it A Celeste make this a little more a general sericterium by Isak probe is there Every committee was by people's appeared in this county 5 . 4 0 But how I recognize that some sir I await a selected group of ah Si Sa from a group for setasides and well I'd make ends are slots here so there Our slot Or beans or boxes or a job assignments order comedy waste I choose the 1st object researchers for and right and to choose a side is without replacement because I want to get artist
1:28:28
A I should say without replaced our and west to notice goes up to Here I'm counting this way 1 2 3 of our to privacy summer Right and my eyes are plus 1 Or you could think of that is being body and minus quantity are minus 1 because some of his suit would start kinder 1 2 3 daughters this number is 1 lesson this 1 the numbers 1 arsonists years are December had subtracting ask me 1 less are what's Wallace arts are minus 1 thing so multiply these numbers together but then we're going to do our mortgages I am not interested in your Anderson isn't having which Elements are there not what were then every permutation of the same group I think he did a pretty through primitive from additional give me the same group Army Times of Iowa over counted when I do this by factorial times I'd like to divide this summer by our factorial thing now Something you've been doing here on this Looks like a frustrated and factorial frustrated wife doesn't get all the way down to 1 stocks What if I put 25 made it have made it in fact I still want this to be equal I have this Pfefferle here I threw in something what I The fight Numbers That start just below that's what starts just below the and miners are within minus are minus 1 always onto 1 so here to abide by and miners are part of our ways to select a are objects without replacement move when the the objects as this here his number you compute baby was by hand or with a calculator be lowered large There are approximations for this for big and written will see later but this will help us for example do the following computation But What's a problem of House improper Arrange Kansas supply cards This probability It is the number of 5 card hands have their down there were ways to get off a lot because this is a problem in a sample space for all funds are equally likely any 5 car hand is as likely as any other 5 cars so they computer probabilities we just need to compute Sighs of the sample space and the size of our events and take the ratio so let's try denominator 1st comedy ways can you give a 5 card and from a deck of 52 is on the board helps to do That's Where are areas 5 minutes 50 by the way there is another Waiter right this would put up Prentice is big enough stolonifer And there's another 1 c event are amusing about this sometimes This is in words we say and choose are and so the denominator here we know it is 52 choose 5 that's a pretty big numbers Where what the numerator what's applause again it's us 3 1 numerical value and tool of another America life so we have to tool tasks or to cast reform refers to pick our numerical value for the river kind and 1st the 3 of the kind in middle these 2 a Caesar Ordered excited triplet here we can distinguish it from somebody ways are there to get 3 of the same Only ways earlier tripled from a deck of 52 Now 13 different values like a king queen For now sewer firstly suck the value 13 visited ace king queen Jack and onto to 13 . not once you've done that get to next rose to Bozoo you're kind Holly Way surrogate 3 2 from Decker 52 vomiting there in the back from those yet to pick 3 talked about right hungry ways a 5th from 4 things yet it bake Pretext for 2 3 numbers easily duped Our if depending 3 things that means 1 waited as they want to leave out So from 4 things are probably ways at once
1:36:16
Similar chocolate strawberry raspberry ME ways Others which is 1 of for Frank energy use this formula down here you get the same result of 8 now so that were wasted 3 of a kind would be the productivity sings 13 times Frank 1st formed task near Nunavut task wave number waste form to get both fast accomplishes 13 times for us they multiplication principal M times and they now let's get value for the basket pair and potty area performed 2 operations 1st pick the value for the their and then pick 3 particular Cards with that nite so now next week at the number of ways that do number waste to select a value for the pair For that being 12th wife O'Leary did the 1st experiment which took that Balyarta consideration like his we have already cards of the that value 1 and we need to so we can use that more of their 12 And given the alleyways 2 cars of a given nite Thanks so each Value free therefore cards in that value humming away so the 2 cards for its fortunes to what would have been carried out Maybe we could do it this way the starving for a reader to forward this being for minors to pass 1 is right or wise to eschew through 1 3 so skyward for ever 3 of his 4 times 3 over through or So that relates to get the 2 of a kind and would be hard to get the pair Would be 12 times 6 Pentagon that's Multiplication principal pursued the value of a given value would the 2 cards that nobody waste Scalia false firstly that verifying to a kind Hardy Phone number ways not by Numbers together so we modify or although this way 13 times 12 times our Porter therefore and that's probably But do 1 more votes Do the probability of a which was a straight or lost Yes straight or philosophy Boss back But let's do much Let's do problem of flush its diversity of that means all the cards are the same suit and hopes is that What they sell homily ways as get a flash well what's the 1st task is to make a suit like states or hearts or diamonds are summers either for now wants you pick the suit picked 5 cards in that suit army cards are in suit 13 Army Reserve 5 from 13 13 Jews 5 3 K So it's winds false or a vast while I do here is barely Graham Watson writer here with 4 times 13 choose 5 every 13 times 12 times 11 down to say it's over Vote of On the oneyear 5 drawing and 8 4 injuries are its 13 factor over 13 months by case now I rewrite this is 4 times 13 times 12 times 11 times 10 times 9 over 5 . 4 0 what happened here I canceled the doctoral with their product of 1 of the 8 thing now what well this 5 factorial has a 5 4 3 2 1 maybe of 5 times that too Can cancel the 10 that is what 4 times 3 and the fourtime streaking cancel 12 Soviet 13 times 11 times 9 times for here that's what this new becomes where but that no rarely so which is bigger follows a forum both sides of the border covers the same 4 years they which is there while force canceled the 13th canceled its is 99 acts 72 so the problem of 4 House less probably a very Flush so false bids for part of so I hope this convince you that probably
00:00
Meeting/Interview
Mathematics
Multiplication sign
Real number
Sheaf (mathematics)
Right angle
Set (mathematics)
Open set
Resultant
Social class
05:29
Statistics
Standard deviation
Vapor barrier
Observational study
Multiplication sign
Student's ttest
Drop (liquid)
Rule of inference
Social class
10:40
Area
Statistics
Greatest element
Spacetime
Observational study
Forcing (mathematics)
Firstorder logic
Multiplication sign
Element (mathematics)
Sampling (statistics)
Multilateration
Mass
Mathematical model
Mathematics
Manysorted logic
Natural number
Moving average
Field (mathematics)
Social class
17:10
Trail
Statistical hypothesis testing
Multiplication
Spacetime
Multiplication sign
Sampling (statistics)
Infinity
Average
Weight
Number
Lecture/Conference
Natural number
Negative number
Field (mathematics)
Hydraulic jump
23:43
Axiom of choice
Point (geometry)
Free group
State of matter
Correspondence (mathematics)
Multiplication sign
Blind spot (vehicle)
1 (number)
Insertion loss
Weight
Mereology
Event horizon
Rule of inference
Likelihood function
Subset
Number
Mach's principle
Frequency
Natural number
Length
Condition number
Rhombus
Area
Focus (optics)
Spacetime
Forcing (mathematics)
Element (mathematics)
Moment (mathematics)
Interior (topology)
Sampling (statistics)
3 (number)
String theory
Set (mathematics)
Line (geometry)
Mortality rate
Category of being
Arithmetic mean
Order (biology)
Noise
Right angle
Mathematician
40:36
Complex (psychology)
Spacetime
Lecture/Conference
Sheaf (mathematics)
Sampling (statistics)
Set (mathematics)
Event horizon
Nearring
42:59
Multiplication sign
Equaliser (mathematics)
Mereology
Event horizon
Number
Subset
Summation
Degree (graph theory)
Lecture/Conference
Negative number
Musical ensemble
Cuboid
Subtraction
Condition number
Addition
Spacetime
Process (computing)
Sampling (statistics)
Staff (military)
Set (mathematics)
Principle of maximum entropy
Sign (mathematics)
Category of being
Summation
Event horizon
Oval
Lie group
Right angle
1:01:48
Point (geometry)
Axiom of choice
Principal ideal
Standard error
Group action
Multiplication sign
Equaliser (mathematics)
Water vapor
Event horizon
Number
Goodness of fit
Mathematics
Lecture/Conference
Number theory
Physical law
Subtraction
Social class
Area
Statistical hypothesis testing
Spacetime
Process (computing)
Element (mathematics)
Expression
Sampling (statistics)
Infinity
String theory
Mortality rate
Set (mathematics)
Sequence
Element (mathematics)
Flow separation
Mathematics
Category of being
Number
Summation
Order (biology)
Right angle
Object (grammar)
1:20:36
Axiom of choice
Metre
Observational study
Multiplication sign
Range (statistics)
Drop (liquid)
8 (number)
Number
Permutation
4 (number)
Lecture/Conference
Cuboid
Selectivity (electronic)
Position operator
Covering space
Process (computing)
Forcing (mathematics)
Sampling (statistics)
Division (mathematics)
Set (mathematics)
Local Group
Field extension
Factory (trading post)
Order (biology)
Hill differential equation
Right angle
Object (grammar)
1:28:25
Principal ideal
Divisor
State of matter
Multiplication sign
Similarity (geometry)
Frustration
Mereology
Event horizon
Permutation
Number
Fraction (mathematics)
MKS system of units
Lecture/Conference
Hardy space
Wellformed formula
Operator (mathematics)
Moving average
Wavenumber
Rhombus
Area
Multiplication
Spacetime
12 (number)
Product (category theory)
Sine
Forcing (mathematics)
Element (mathematics)
Sampling (statistics)
Euler angles
Bilinear form
Local Group
Approximation
Maxima and minima
Calculation
Numeral (linguistics)
Voting
Factory (trading post)
Right angle
Object (grammar)
5 (number)
Limit of a function
Resultant
Nearring