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April 16 - 2nd period

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going to get started problem number two this can be written as 7 p 2 hopefully is 7 factorial / 5 factorial which would be 7 * 6 other way of writing 7 * 6 formula 72 this is the one that's not like the others because the other three are all just different ways of writing the same thing 7c 2 and 75 at the same the course this is the formula for when you do 72 call number 74 shipping super easy if you understood it when you read it Cartier they ask how many different ways can you choose marbles marbles in the bag so if you're going to grab three marbles from a bag of three marble obviously there's only one way of doing that of course the illustration they have on the route 74 is illustration of all the different orders which you could choose marbles so course there's six different ways you could have known 61800 stration because 3 factorial in 4 how many ways can you arrange 3 items and 3 factorial is 6 or 6 ways of arranging three items are there any questions on homework so I have a question prom number 49 I think 49 comes out and use the word you're asking the question number 49 I want you to guess what word in this sentence is the most important in the sentence you must correctly select 6 numbers each an integer from 0 to 49 the order is not important just about yeah yeah that's true I mean order not important is the most important thing here that tells you to use Sea I was thinking the word select it's really important cuz slack is like synonymous with choose so when you see a word like select or Choose You tendo use see but I noticed 50 they do use the word slack but then they say order is important so that overrides that use the word select so if you're doing over 50 you have to use p I can't hear you just going to use see there is 50 numbers to choose from because you're including the number zero so it's just 56 50 C6 I'm sure set alarms number that I would use a calculator to calculate 50 ft 6 that you have to be correct 50 six ways of choosing numbers this is like a lottery actually said it's a lottery so you have a 150cc 6 ways of winning I mean your chances are only one out of 56 is 15890700 next question I have is 77b it says how many different 5 Card hands have all five cards of a single suit let's start with how many 5 cards hands are there 2 cards in a deck to create a hand the reason they call it a hand is and has five fingers that's a hand in poker it's always five cards if you are playing a poker game where you're given seven cards the cards make your hand so if you're playing a poker game best cards to make the hand hand is always five cards in poker no matter what game you're playing so how many hands are there in poker I don't remember this number but I think it's kind of interesting number so that it's a remember so 52 shoes so there are about 2.6 million hands in poker 2.6 million chameleon hands we're looking for the hands we're all five cards are the single suit let's say they're all hearts there's 13 hearts in a deck and you have to choose five of them to make an all heart hand so this is the number of that you can create a hand in poker that's all hearts how many ways can you create a hanbat is all states the same because there's 30 how many different ways can you create a hand at all diamonds how many ways can you create a hand that's all clubs Brothers four four of these These are the number of hands that are all clubs all Spades or diamonds all hearts I can go ahead and write 13 C 5 + 25 + 13 C 5 but you know why do that just four times that that's it that's how you get the answer what does it say in the book it says number 77 Oasis how many it's not a probability question sorry about that I thought it was a probability question cuz I was thinking about number 49 there just asking this so I yeah whoops that you should just go on for 13 and CR V 5148 I'm sorry about this bottom part here you know whether something interesting to learn from this bottom if you are dealt five cards right now I got you shuffle a deck and I just deal 5 cards to you for those of you who know what I flush is a flush is when you get all the same suit this is the number of different flashes there are these are the number of different hands that are flushes 2598960 hands so someone U V car is right now off the top play well Chef Shuffle deck this would be the probability I know this wasn't part of the problem so you don't cross that out lightly this would be the probability that you be dealt a flush not a big probability mascara Percy's played poker before you probably think wait I've gotten flashes lot because there's something about the game that allows you to use more than just buy Cards your first day out some new cards try to make a 7 cards to work with so that really improve your chances you can see that the chance of just being gout right off the top of the deck is super small looks like it's about 1 in 500 wondering how do I get that one 500 this is not 5,000 this is about twenty-five hundred thousand something like 5 over 2500 drop all these numbers what is that that's about 1 in 500 when you Shuffle up a deck about one out of every 500 shuffles is going to have the same suit on the top five of the deck I sent way too much time on that sorry about that I got to go on next question question number 69 Kayana phone number 69 by one of the problems is the answer the butt is weird that's probably what led to some people being confused by number 69 if you take a good look hey at the diagram they have their show you what I wrote in my book I was talking about this problem my previous class is what I wrote in my book so if you look at this diagram that's not hard to see a big enough screen can see this there are five vertices in this diagram five vertices how to draw a cell connects two vertices what you need to do is you need to First choose to vertices to connect how many different ways can you choose two vertices if you are given five vertices 5 choose 2 comes out to 10 so what that means is if I have five vertices there are 10 different ways to draw a segment connecting to a vertice if you look carefully there are actually 10 segments in this diagram why are there 10 segments in this diagram because five 2 is 10 why is five choose to being 10 the key to understanding this because when you are connecting two vertices the order in which you select vertices to connect is not important appropriate formula because order doesn't matter if I select this vertex and then this text to connect about the same as choosing this vertex and then this vertex the order in which I slept the two vertices to connect not important when order doesn't matter you use C and evidence of this is go ahead and count how many segments CC sure enough there are 10 segments the confusion here is is they asked about not every segment up here is a diagonal the segments on the outside are sides of the Pentagon a 5c to include diagonals and sides if I want this only to be diagonal I need to subtract the number of sides number of sides will always be equal to the number of vertices number for 10 - 5 is there are five diagonal in here so how do I answer 69a use the combination formula to write an expression for the number of diagonals in an N sided polygon set up saying 5 I'm going to go and see two segments but I only want diagonal so I need to subtract the sides and that's it so the correct answer to part A should be NC 2 - 2 my text look I don't know why they have NC and -2 - because I explained to you guys in my nose that like a hundred see two and a half c98 the same thing well NC to + NC + -2 are actually the same thing I just don't understand why they would think about what I've been talking about you know what it what are we doing here we're drawing segments by choosing two vertices to connect two total sense to me so that's why I think you should how do you do Part B so writing down the answer to a it's NC to -10 for Part B they want to simplify this is like np2 / 2 factorial is remember the only is that you have to do the extra division by this number factor so what is NP to factorial you supposed to stop after one two numbers so I went too far here to get rid of those are you can if you're confused think about it 10 P2 would be 10 * 9 100 G to the a hundred times 99 the end times one less than in divided by 2 factorial is 2 - 10 get a common denominator by multiplying top and bottom by to hear so now I get N squared - + - 2 and over to the end squared minus 3 and over to the answering the book not the end I mean you don't really need to do that but that's how you get the answer to match up what's in the butt are there any the number 69 can I use one more question I have the number 73 I'm pretty sure in the app on prime number 73 I didn't have time in between my tooth the song my previous class asked about number 73 and I didn't have time to look at what they did in the app but my guess is that they used like one of the formulas from the section I came up paymaster that I hope works that doesn't involve using a formula from this section turns out that I I'm giving you the wrong answer I apologize I I think this makes so this is how I explain it and I think this is going to get the right answer doesn't give me answer that's in the back of the book I don't know what to say cuz I'm pretty confident that this is correct I do not have time to verify the dancer in the book is correct so this is a potluck where 10 people Mission are in a list that people are supposed to bring a knife and the question choosing from a list of 10 items what's the probability of a potluck disaster so what is a potluck disaster coming with dishes a potluck disaster would be at least two of them bringing the same item imagine six people are coming and two people bring dip be a serious disaster no one brings chips to people bring dip that's a potluck disaster it what's the probability of at least two bring the same item now difficulty with this problem is is that there is a large variety of potluck disasters you can have two people bring the same item you can have three people bring the same item you could have two people bringing dip and two people you know bringing chicken can one person being bring drinks in one person bringing the I don't know vegetables whatever that's considered disaster because again at least two people are bringing there are so many different ways of a disaster that what you really need you on this problem is flip the problem on its head and instead of trying to figure out different disasters you need to do one minus the probability of potluck perfection that's way easier than trying to figure out all the different ways are you going to have a disaster sad figure out how many ways and what's the probability of potluck Perfection and then just saying one minus the probability of potluck Perfection is the answer what's potluck perception no two people bring the same item see there's so many different ways of a disaster but there's only one way of perfection it's my easier to calculate the probability of perfection so that's what we do what would potluck 4 section look like the first person comes in with their dish nothing bad can happen right because bring any of the 10 items to the first person to bring anything the next person better bring something different than this person what's the chance that the next person bring something different 9 out of 10 what's the chance of the third person bring something different well as long as that third person doesn't bring this person's item or this person's item we're okay 10 chance of that happening then continue down the line tell you right now in this particular scenario I don't think the chance is very good that you're going to get potluck Perfection this is going to be a pretty small fraction and majority of the time you're going to have a potluck disaster cuz you didn't Ino assign people in item to bring answering my book I don't know if it's correct that I'm a copy of it hopefully what you get here is this cuz I am very confident this is the correct answer this is the answer in my book can see the great majority of the time you're going to have a disaster probability of at least two bring the same item pretty high Round Table round 80% since that's my explanation notice I didn't you see at all on this I just used the idea that each time it's less and less like disaster and bring an item that hasn't been brought yet this shows you the importance from first half of the chapter the importance of complementary events the purpose of complementary events is absolutely huge cuz it's a one second shear the importance of complementary events as a concept is huge because makes a problem that could be incredibly complicated into something that's relatively simple I hope you understand the difficulty all the different ways that at least two people can bring the same item is there's just too many of these too difficult of a problem are we going to go on to the last section the new material right now Toulon going over all that show the last topic the chapter call binomial distribution start with an example so I can teach you some vocabulary is let X be a random variable that represents the number of heads when 4 coins are tossed according to the textbook a random variable is a variable whose value is determined by the outcomes of a probability experiment can a probability experiment in this problem is 4 coins are tossed that's the probability experiment and the value of the random variable is determined by text you get when 4 coins are tossed first thing you should understand is that a random variable in one of these problems is going to have a very restricted domain using axe here and you might look at ex and say well excuse me anyting X is the universal variable that we use in algebra where to be a decimal to be a rational and be all sorts of it not on these problems if x is a random variable and one of these probability experiments it is going its value is going to be restricted by the nature of the experiment want to see if anyone can call me right now chat what are the only values that X can have in this problem just type in chat only values that X can have problem otherwise what I'm asking you to type in is the domain of X what are the only values of x you're not right 0 1 2 3 4 incredibly small domain of a value of a variable like x x to be so get used to that talk about random variables in this section the domains of these variables going to be very very small in this case the domain is only 5 numbers if you're confused why is it's your one two three four 4 coins what's the fewest number of heads you can get zero What's the most number of heads you can get for can you get any number of heads in between 0 and 4 absolutely we make a table and a histogram showing the probability distribution of X what does that mean we're going to show hey what the chances are that's w'y the chances that we get one head different possibilities think about which ones are they most likely and which ones are the least likely I hope you realize that the least likely would be 0 heads and foreheads those would be incredibly unlikely best possible and why would they be incredibly unlikely do not do this on any of the homework but here's a list look at this list these are all the different outcomes if you throw for coins like right now you can get for coins can the first coin could be heads or tails II coin to be heads or tails they're going to be heads or tails the 4th going to be heads or tails they could be all heads they could be all tails three heads several possibilities of all the different outcomes for the four coins which of the least likely what's the most likely even balance we keep in mind if we have a situation we had a situation I were to throw for coins in the air right now and ask you know okay what do you think's going to happen best two heads to tails but if you look carefully at this even though that the best guess as to what to have what what happened majority of the time because there are 16 different outcomes listed here and only 6 out of 16 which is 3/8 3/8 I have two heads and 2 Tails so even though a plurality of the outcomes have two heads and 2 Tails it's not a majority so if you know we had this as a you know you go to someone saying I'm going to flip four coins right now on a flip four coins right now and I want to make a bet with you gay and I'm going to allow you to choose whatever outcome you want and we can even money back based on that outcome horse that person's going to possibly if they don't understand math very well go for this because I get to choose whatever they want to go out shoot most likely outcome is who heads to tails maybe right it is the most likely outcome I would still be a terrible best for them because the chance of it coming up two heads and 2 tails is only 3 out of 8 number 6 out of 16 Sobe actually end up being a terrible bet even though they feel like they are in control well it is the best choice they could possibly make still not going to give them a fair chance of getting their money back or I should say doubling money okay this is what the table looks like you do not need to make this to create the tables on tonight somewhere so how are you going to come up with these numbers here is work so in the chat can someone tell me how can you get these numbers without 14641 those are the important numbers in the table 14641 Anthony got a ride and the answer is Pascal's triangle 14641 is just the 4th row of Pascal's triangle when you do these probability experiment on tonight's homework look at how many trials there are this case search for trouser flipping 4 coins because there's four trials you're flipping four coins you're going to be used in the fourth row of Pascal's triangle if you were doing a problem that involved flip a coin 5 times it doesn't have to be flipping a coin any kind of binomial experiment with a binomial experiment is an experiment where there's only two possible outcomes so I don't want you to think it's only about slipping point it could be a winning the game versus losing game it should be about rolling a 6 not rolling a 6 I want to buy that last one okay when you have a die there's a six-sided die a six-sided die in six sides to it so how am I going to do a binomial experiment with a die cuz I just said a binomial experiment is an experiment that only has two possible how am I going to do a binomial experiment when they die it has six sides to it I want to see experiment is do I roll a 6 or do I not real with 6 that is just two possible outcomes that is a binomial experiment either you get a six or you don't get it fixed that you'll notice that in that example probability of those two outcomes is completely different the probability of rolling a 6 is only one out of six the probability of not rolling 6 is 5 out of 6 that's okay I'm these problems Katie's binomial experiments that we do we are not necessarily going to be all like flipping a coin where there's a 50-50 chance of each outcome listen to me when I said this could be about winning a game versus losing again you know if you're part of a team that's really really good better than 60% chance of rain 40% chance if that's still a binomial experiment as long as there's two outcomes is a binomial experiment anyways and these problems you got to have to figure out first how many trials there are in the problem is there are four trials cuz you're asking for coin so you can use the 4th row of Pascal's triangle if you are doing a problem that involves 5 trials for example of a problem involving that you are part of going to go ahead and play five games against well in that case to make the table you can use the 5th row of Pascal's triangle now remember last session I told you that each row of Pascal's triangle can be found without actual figuring out all the previous Rose so if you're doing a problem involving five trials the way you would get these numbers Hugo 5c 05 C15 c25 c35 I see five you can get that row Pascal's triangle without actually you know doing all the previous res in this problem so how do I get to last row when you flip a coin 4 times slips for different points coin has a problem has a chance of being a head or tail so the number of different ways that you can slip for coins is 16 because it's 2 * 2 * 2 * 2 so that's where I get to 60 course I can also say I got the 16 from this but I just said on your homework please do not do this this with just for explanation purposes on your homework when you get a problem like this this is what you do you write appropriate row of Pascal's triangle and then you write the probabilities by taking the numbers in this route K and dividing by the number case of flipping a coin 4 times There are 16 possible outcomes cuz to the 4th and 16th here's a histogram of these results so it's get the history can you take the last row of the table and you wrap it when you are doing binomial experiments the expanse I have an equal likelihood of success and failure in the case of flipping a coin and seeing whether it's heads you're going to have equal chance of six failure there's a 1/2 chance of it succeeding and getting ahead a 1/2 chance of getting a Tails when you have that equal likelihood of success the histogram will always looks to metric will always look symmetric when you have an even chance of success and failure the histogram will look symmetric just keep that in mind show you something to kind of contradicts that seeing this problem right here it says there's a 33% chance of success I'm not going to just trust me that's a chance of success look at the histogram it's not symmetric what is it weighted it's weighted towards less success see how the bigger bars KR in the smaller numbers it's weighted towards last success you actually call this skewed right this is called skewed right histogram would be skewed right with a small percentage this was a big percentage high percentage of success then the big bars with skewed left answering a couple questions about this what is the probability I'm sorry what is the most likely outcome obviously already talked about this example most likely outcome is two heads now it's not a majority of outcomes a plurality but not a majority but it is definitely the most likely outcome two heads and ice 3/8 chance of that happening what other question what is the probability that at least two heads are tossed at least two heads are tossed with six outcomes have two heads for outcomes have rehab Malcolm where you get at least two heads so that's 11 out of 16 so there is a 11 out of 16 Champs these questions how you can look at the table or the histogram and answer questions if you need it last thing formula for these problems so after you pass the beginning of the homework actually that it's not just the beginning there's other parts the homework we also have to do a histogram but it's a problem that doesn't require you coming up with a history which means you don't have to come not with a multitude of probabilities they only ask for one answer when you get in trouble when they only ask for one answer not a whole bunch of probabilities not an entire histogram of probability just one problem this is the formula you're going to eat the first of all this only applies to binomial experiments what does that mean in experiment when there's only two possible outcomes success or failure in the case of flipping coins that would be heads tails in the case of playing a game win loss and a case of rolling a die and getting a six I6 not getting six what is my last two examples the two probabilities the probability of success in the probability of failure are not going to be 1/2 like for example rolling a 6 is a 1 out of 6 chance failure is 5 out of 6 in the case of playing a game and winning I could win what's a 60% of the time the point six would be my probability of success while I lose 40% weight for would be my probability of failure if you're going to play a certain number of games how many use that as an example let's say you were going to play five games what's the probability of winning three times where you at choose three and I want to win three times I would go 5 C3 probability of winning points 6 raised to the Third x points for which read a probability of losing raised to the second in the state that so many people make when they get a problem like the one I just described involving here's some steak and so many people make a say let's say I have a 60% chance of winning again 60% chance of winning and there's five games and I want the probability of winning exactly 3 * here's the mistake people make beagle I guess they go okay I want to win three times so is a point six chance that I'm going to win when when and then lose and then lose last three way and two losses so this should be my probability think about why that is wrong the reason this is Ron the reason point 6 raised to the third times 4 squared the reason that is wrong and is because this isn't the only way to win three times this is winning the first three games then went then the last you could lose the first to win the first game there are all sorts of different ways of winning exactly how many different ways are there choose three that's the number of different way that you can shoot that you can win three times why does this formula work why is this going to correctly tell me the number of ways because in a game Series right where you have these are the game numbers so you have games 1 2 3 4 and 5 how many different ways are there to win three games well that's going to always be equal to the number of ways that you can choose three out of these five games to win like for example I could win the first 3 games I could win the first third and fifth game I could win the third fourth and siskin K5 C3 is going to give me all the different ways that I can grab three of these game numbers to win could be this game this game in this game that I went so that's the reason you have to have this extra part right here because there are lots and lots of different ways okay for you to have any particular number of successes so here's an example using kind of similar numbers here I have a situation where it's 62% of adults you ask 5 adults whether they have visited a dentist in the past year what is the probability that exactly two have been have visited today here's the setup real simple there's five trials cuz you're asking five different people you want to have exactly two successes okay this it would be the probability of success success failure failure failure if you write this is your answer you'll be wrong because you're only include success success failure failure failure the reality is is there is lots of different ways of having two successes it doesn't have to be the first to adults that say yes it could be the first the last that says yes there are this many ways that you can have two successes there's 10 different ways of having two successes so take the probability of success success failure failure failure x 10 because there's 10 different ways of achieving two successes in this is what you get but this first part is not Siri because there's always in a binomial experiment multiple ways of the shaving your end goal in this case my end goal is to have exactly two adults say yes to me when I ask them do you practice good oral hygiene did you visit the dentist in the past year okay I want to have exactly two successes 10 ways of having two successes probability of having any success. This is 10 different ways of achieving it so that's it sorry random over I'll stop there

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April 16 - 2nd period

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