 So let's start lesson one y'all there The following says warm up the following sample probability questions were asked in junior high math. I need the chatter to stop Hey, we're starting that corner. Thank you. I Don't know. What did I say page 423? I think I said or 424 something like that Okay A fair die is rolled by the way English vocabulary to dice one die I'll often say one dice because it's kind of become an idiom that's part of our language But if you're reading a book and they talk about a die, that is the singular of dice We'll be grammatically correct in our typing or on the exams a fair die is rolled What's the probability of rolling a one now? Intuitively, you know the answer to this in fact basic probability We have a reasonably good intuition on but anything beyond really basic. We're lousy on Hence Vegas hence Horoscopes hence Gamma all the super all the patterns we try and find we're really lousy at it. So basics one out of six a Circle is divided into four equal sectors labeled clubs diamonds hearts and spades When it's spun, what's the probability it lands on hearts? one out of four two coins are thrown and The number of heads is counted What's the probability of getting two heads when you throw two coins? It's a bit of a trickier question. Let's find out It seems to me that you could get heads on the first heads on the second Tails on the first heads on the second tails on the first Tails on the second am I missing anything or me? I'm missing one heads on the first Tails on the second have I missed any possibilities? By the way, can you see the possibility for permutations and combinations to start showing up if I treat this a little later? Yes, that's why we always do combinations before we do probabilities Yeah, I think it's one out of four looks like and one of the first things We're gonna start to ask ourselves is there a way to get this without counting But a short rule we have in probability is this if you can count it you can calculate it when in doubt Write them out count them up Now that works great for small questions. It doesn't work great for picking five cards or anything like that too many possibilities Terminology probability theory deals with the mathematics of chance or Prediction we're going to use the following terminology a trial is any operation whose outcome cannot be predicted with certainty For example flipping a coin Rolling a dice Dropping a ball is not a trial because I know it's gonna hit the ground What are the odds of the ball hitting the ground? 100% long as there's no desk or something in the way But any event where there is some uncertainty about more than one outcome called a trial an Experiment consists of actually doing one or more trials We're gonna use those fairly interchangeably though an Outcome is the result of carrying out an experiment flip a coin get a heads the outcome is heads Roll three dice the outcome is a six a six and a four the sample space Often used with a capital letter s is the set of all possible outcomes Here is the sample space for flipping two coins there are no other outcomes beyond those four and event is A subset of the sample space for example this question was asking about the event two heads from this sample space heads an even number could be an event if you were rolling a dice and Here's our first official definition of probability This one's pretty intuitive. I'll you know highlight it But I think most of you naturally understood this when you guys gave me the one out of six answer If an experiment has a set of equally likely possible outcomes Then the probability of a particular event a is given by the formula We're going to use some notation gen lots of shorthand instead of writing the word probability You know what letter we're going to use a capital letter what? P and then we're going to do an open bracket and the event is going to go in the brackets in this case the event a and It's the number of outcomes favorable to a Divided by the total number of possible outcomes probability of heads one of a two Probability of getting a five when you roll a die one out of six. So this is going to be our notation Pretty good pretty short What's the probability of something being impossible if something's impossible? What's its probability? if it's impossible we would say P of x is zero for example, what's the probability of rolling a nine on a six-sided standard die? You get a nine zero Okay, that doesn't sound very profound. It's more profound than you think oh and if something is guaranteed to occur We say that it has a probability of one 100% if you want to think about it as a percentage for example, what's the probability of rolling a natural number less than seven? What are the natural numbers less than seven six five four three two and one? What's the probability of rolling a one or two or a three or a four or five or a six on a die? Guaranteed what's the probability of rolling a heads or tails? guaranteed Okay So guaranteed mathematically one that gives us our first built-in error check gives us our first built-in error check Rio for any event a The odds of it occurring have to be between zero and one if you get an answer bigger than one you've messed up You've done something wrong on your calculator. You've forgotten brackets. You've done something if You get a negative answer you messed up All probabilities are either impossible or guaranteed or anything in between Often we're interested in the odds of something not occurring And we have an abbreviation or a shorthand for the word not if we draw a Horizontal bar above something we read that as not a when I write this this is actually the probability of not a occurring and We call it The complement of a oh and there's a handy dandy little equation The probability of not a or not a is equal to one minus the probability of a That's on your formula sheet, but it's worth having in your brain Let me give you an example Supposing they tell you the probability that it's gonna rain tomorrow is one out of three What's the probability that it's not gonna rain tomorrow then? Two out of three That's gonna become handy because Dylan you're gonna find for a lot of questions It's way easier to find the complement than it is to actually find the answer to the question that you're looking for But then just go one minus the complement. You're done. We're gonna use that all the time So this formula here is on your formula sheet But Jen I'm gonna say you'll memorize it naturally because it's kind of intuitive for each of the examples It says state the span sample space Which outcomes are favorable? Whether the outcomes are equally likely and the probability of the event So a fair die is rolled. What's the sample space for a fair die and? Traditionally we use the squiggly brackets because it's a set what are the outcomes for a fair die one comma two comma three comma four comma five comma six Now the event it asks is what's the probability of rolling a one? Okay, I think that's and I'm just gonna circle it that event right there What is the probability of rolling a one we all know this one? If it's a fair die, what is the probability? 106 oh Are all these outcomes? I should have answered part three are all these outcomes equally likely is it this? Yeah, that's gonna be a key right now right now. We can't handle Non-equally likely outcomes. We will in a couple of days. We'll add a clever trick a Circular spinner is divided into four equal second sectors. We did this one already We said the answer was one out of four Two coins are thrown and the number of heads is counted. What's the probability of obtaining two heads? We did this one already as well. We listed our sample space. We'll do this one over here though We said the sample space was Now it depends how you list the sample space if you list the sample space as heads heads tails tails heads tails You can get two heads one head or no heads Are all of these equally likely the answer is no Because you can get that two different ways And so we would either have to list it the way that I did earlier by calling it heads one heads two tails One tails two heads one tails two tails one heads two and then they're all equally likely or Preview of coming attractions So not equally Likely in fact if I was doing this question. I wouldn't list my sample space this way I would list my sample space as The way I did before heads one heads two heads one tails two tails one heads two and What have I missed? Tails one Tails two and there I think those are all equally likely Because each of them will only occur once and then I could get the answer of One out of four not one out of three probability of heads heads Is one out of four So in part B says why can't you use the number of outcomes divided by the total number of Outcomes you can't for the blue set here because they're not equally likely next page compound events Events formed from repeated trials or from a combination of simple events are called compound events and Often a table a chart or a tree diagram is Useful in determining the sample space the two we're going to use the most are a tree diagram and a chart In fact, we'll use a tree diagram Nick 90% of the time We'll use a chart a small amount of the time and you'll figure out pretty quickly when we use want when we use what Says this consider an experiment of rolling three equally spaced So I've rolling an angle try that again consider an experiment of rolling an equally spaced Triangular spinner number one to three. There's a three somewhere and tossing two coins Says draw a tree diagram to show all of the outcomes So the first thing we can do is we can roll the spinner. There is Three possible outcomes a one or a two or a three then You can flip two coins What are all the possible outcomes when you flip two coins? heads heads heads tails Tails tails heads heads tails tails tails heads heads tails tails tails Now this is using the non equally likely outcomes approach. I don't like that I want instead of three options here. We said technically in the sample space How many sample how many elements are there in the sample space? For do you know your rights? I'm gonna go like this heads one heads two one two one two One two one two one two one two one two And then I'm gonna add one more branch to each one of these if I can. I'm gonna say, okay Here is which one am I missing? tails one heads two Tails one heads two Tails one heads two Tails one heads two Now why is that so handy? Oh, it says list the sample space at the end of the tree I don't want to list them all Instead, I can show them to you Jan one sample space is one tails heads Another one is one heads heads another outcome is one heads tails Another outcome is one tails tails You can walk down any branch and if you walk down every single branch systematically You'll have listed the sample space, but they're contained in the tree I like that better because it's way less writing Dina, can you see one possible outcome is a three tails tails as well as a three tails heads a three heads tails and a three heads heads They're all there How many outcomes are there count the number of branches? Brendan is there a way that we could have used the fundamental counting principle to figure that out? Well, it would have been yeah three numbers times four outcomes the tricky part would have been ahead of time spotting There was four outcomes. You probably had to make a list Are all the outcomes equally likely the way I've drawn it here? Yes So now I can answer these questions for part D the probability of getting a three comma H H is How many outcomes were there grand total? 12 How many outcomes are favorable? How many branches end in a go with a three heads heads? Just one what about the probability of getting a prime number and Exactly one tail Which of these numbers are prime One is not prime by definition primes are any number greater than one There's two prime Yeah, two and three are both prime numbers because they only go into the only divisors They have are themselves and one one is not considered prime. So Here here and exactly one tail I find what helps me a lot if I've done a tree is to simply change colors or not and put checkmarks under Satisfactory outcomes. Here's a two and exactly one tail. Here's a two and exactly one tail Here's a prime and exactly one tail. Here's a prime and exactly one tail by doing that It's very easy for me to spot the number of favorable outcomes Oh and by writing them below the tree Madison if I need to use this tree again I just pull up my eraser erase those checkmarks and start over or if it's pen scribble it out and start over But I would think well, how many favorable outcomes are there? four out of I know that's one third Doug we're not going to reduce probabilities Because 99% of the time you're gonna want to common denominator anyways now They will reduce them in their final answer. So I will trust that you still remember your fraction button math enter enter Now this one, you know is one third but for the more complicated ones like the card questions that are over a question in the thousands If they want me to reduce the fraction that's gonna I'm just gonna go like this math enter enter and that's gonna write it as a lowest term tree When will I use a chart? Any time I'm dealing with two dice You know why how many outcomes are there using the fundamental accounting principle with two dice How many branches would my tree have if I use a tree then? 36 no Trees are for smaller samples, but they're better in a lot of ways I use a chart for a dice and in fact what I would do here We have a red die and a blue die. They've started this chart for me But often I'll just freehand it it would I would write down red blue I would write numbers one through six one through six and then I would just add a line like that and like that This is one one one two One three fill in one four one five one six It takes about ten seconds to do a dice chart, but it's so handy. It's well worth the while two one two two two three two four two five two six three one three two three three Five three four is already there 36 says show all the possible outcomes in the array done How many points are in the sample space? What did we say by using the fundamental accounting principle? 36 says List the event the same number appears on both dice as a subset of The sample space by the way, what do we call it when we're playing a game and we get the same number? You roll it in Monopoly Huh doubles do we not still call it doubles? No, that's only for two ones. That's only for a subset of the subset I agree with you Brandon that that's one possible outcome, but if we're talking the same number that's not snake eyes That's a Brandon Brett caught myself now There's a list of all the doubles. Of course. I ran out of room Okay By the way, if they don't give you this chart and I will ask you a dice question on your test. I Just go like this Freehand one two three four five six one two three four five six One one one two one three you can whip you can write it very fast so What's the probability? That the same number appears on both die or as we like to say when we play Monopoly Brett, what's the probability of doubles? How many outcomes are there grand total? 36 how many favorable outcomes are there? Six and I know that's one out of six and lowest terms who cares What's the probability that a different number appears on each die or a shorter way to write that is what's the probability of? not doubles Don't count use the compliment If six out of 36 is the odds of getting doubles, what are the odds of not getting doubles? 30 out of 36 because they are opposites to each other. There's no overlaps. It's the compliment That's where we use the compliment. It would be way easier if they said what are the odds of not getting doubles I would find the odds of getting doubles because I know there's only six of them count that and then subtract Part of your homework today, but we're gonna move on to lesson two as well number one three four So I think I skipped two. Yes So one skip two three four skip five six is nice eight and 11 Having said that let's go to lesson two Here is where the fundamental meat and potatoes kind of begin here more terminology and we're gonna talk about We're gonna talk about the events a or B and the events a and B We got a bit of a problem here because the word or in math means a little bit different than it does in English so in Mathematics the event a or B is Said to occur if a occurs or B occurs or both occur and That's the difference between math and the English language if I say to Doug Doug I have two chocolate bars. You can have a Snickers bar or a Mars bar. We understand in English Both is not an option in math. It is so in math or means one or the other or both and What oh the event a and B occurs if both events occur simultaneously at the same time And one of the better ways one of the easy ways to visualize this is through something called a Venn diagram So it says this Consider the experiment of noting it of rolling a die and noting the result Let the event a be an even number is thrown Let the event be be an odd number is thrown put the outcomes into a Venn diagram So our outcomes for rolling a die are the numbers one through six which ones would end up in event a If event a is an even number is thrown which ones are here two four six which ones are here one Three five Combine those two together and that is the event rolling a die So it says list the outcomes for the event a can you see them from the Venn diagram? two or Six the event be one Three or five What about the event a or B? So that means one or the other or both? Any numbers that appear in one or the other or both? I think Madison That's all of them now the event a and B means which ones are happening at the same time Are there any numbers that are odd and even at the same time? Empty set zero with a line through it. Don't just put a zero because zero is a number zero with a line through it, okay? C says let N of a I think we're gonna skip straight to D Looking at this Venn diagram. What's the probability of event a I need something out of something? three out of Six I know it's one of two. We're not gonna reduce What's the probability of event B? Also three out of six. What's the probability of events? Oh a or B? Six out of six which is one Right or means one or the other or both odd or even or both there is no both in this case But we're gonna talk about what that means. Oh, what's the probability of a and B? zero Out of six zero Here's what I want you to notice. So here's some more terminology lots of terminology today This one we're gonna put a bit of a star around because this one they will ask you on the other ones You'll learn them naturally this one kids get mixed up. It says this if A and B have no common outcomes No overlap They are Mutually exclusive and I'm gonna put a big circle around there Mutually exclusive means no overlap Black cards and red cards are mutually exclusive Our black cards and spades mutually exclusive No, some spades are black our black cards and Kings mutually exclusive No, some Kings are black Our black cards and hearts mutually exclusive. Yes Okay And you can easily tell an event diagram if they're mutually exclusive because there's no physical overlap Says verify the following rule the probability of a or B equals the probability of a plus the probability of B I think I said to you guys last unit or means add. It does work next page Once again, we're going to roll a die and note the result The event a is an even number is thrown The event B is a multiple of three is thrown Okay Where is what's going to go in a? to a for Can I put the six right there? Why can't I put the six right there? It's also a multiple of three. I need to put it right there because it's both What else is a multiple of three? That's going to go right there because it's not even but it is a multiple of three. What number am I missing? What numbers am I missing? They go Outside there's my whole set so it says list the outcomes for the event a to for or six List the outcomes for the event B to Sorry B. Mr. Dewick look at the right diagram three and six The event a or B two three Four or six one or the other or both you don't write the six twice though even though it appears in both What's the event a and B? six Once again, we're going to skip C. We're going to see if we can go right down to D. So What's the probability of a occurring? three out of six What's the probability of B occurring? two out of six What's the probability of a or B occurring? four out of six What's the probability of a and B occurring? Here's the question Can you use these numbers to get an equation and the shorter answer is yeah, if you want to find the answer to or It's the first one plus the second one minus The overlap See it five minus one Four and that works from the previous page as well if I wanted to find this answer here or It's the first one plus the second one minus the overlap But what was the overlap in this earlier example because they're a mutually exclusive The overlap was zero and this is going to give us our first official probability formula equation So because these have common outcomes, they are not mutually exclusive and This gives us our or equation right here the probability of a or B is the probability of a Plus the probability of B minus the overlap oh and if the overlap is zero They're mutually exclusive and You can use that because the overlap is zero But I really don't do a big song and dance about this one because Jen It's a special sub case of the general equation right there Jen how many terms are there in this equation count them? For that means if you know three you know one if you know three and over the fourth So usually they want you to find that sometimes they'll give you this and this and say find the overlap or Sometimes they'll give you that and that and or that that and that and say find the probability of be any any three gives me four so Here are our key ideas so far Events are said to be mutually exclusive if no overlap in the Venn diagram and Or means add Except we have to subtract any overlap because we'll have counted it twice. I'm gonna put a box around this one Remembering that the upper equation is a special sub case In other words if this is true, then you know, they're mutually exclusive Lots of terminology, but hopefully Doug you're seeing the math so far not too tough. We're counting and I don't even think we need to pull off our shoes yet. We've been counting up to ten at the most I think oh no, we did have a 36 didn't my bad. So example one State whether turn the page if you haven't already state whether the events a and b are mutually exclusive or not a Lot of this you can do with straight common sense what you're asking is is it possible for something to be both of these at the same time? Can you be a face card and a club? What's a face card? Jack Queen or King is it possible to be a club and the Jack Queen King? So these are Not mutually exclusive What about this next one? To dice or thrown is our event our experiment Event a is the dice both show the same value Brett what we call doubles Event B is the total score is 11 Can you have a score of 11 and be showing doubles? example to Which of these events are mutually exclusive? Which of these events can't happen at the same time? Adam Event what an event what? Okay event a and c you can't be a face card and you can't be a jack-queen king and an ace at the same time You're one or the other B and D. Can you be a club and a red card any others? I don't think so So this is Megan the common sense approach to mutual exclusivity and that's if you know enough about the event that you just know There's also a mathematical approach where you don't know what the event is. You just know the probabilities First of all from a Venn diagram here says which ones are mutually exclusive a is mutually exclusive from everything B and D are mutually exclusive Shit That like a and c but a and b and a and a is from everything what I want to do is go to example for here's a mathematical one It says use the following information to determine whether the events a and b are mutually exclusive And they've given me the probability of a the probability of B and or I'm gonna write down our equation We said this the probability of a or B is equal to the probability of a Plus the probability of B minus The overlap a and b this is on your formula sheet You don't need to memorize it. Although many of you will because you're lazy When will this be mutually exclusive? This will be mutually exclusive if the and term ends up working out to what? Zero so let's plug in everything else What's a or B? Seven over twelve one over four Plus one over three is that zero I'm not gonna go fancy. I'm not gonna go all common denominator e hit your calculators out I'm gonna go one quarter plus one third I'm gonna hit enter and then I'm gonna hit math and or and or You know what one quarter plus one third works out to in lowest terms seven twelfths oh So What does this term work out to? Does work out to zero? Therefore P of a and b equals zero so mutually Exclusive no overlap and you'll notice I was able to do that deal even though I have no idea what events those are No clue Could be betting on something. I don't know could be the odds of your car starting I don't know a Venn diagram is very very useful. In fact, I would say this anytime they give me two outcomes and Percentages I go Venn diagram Example five a grade nine class was surveyed to find out whether they did math homework or English homework last night so Looking at this and assuming we're not talking about this class What percent did math homework and don't say 47 because 47 is wrong? What percent did math homework not 47? Evan what or said what the entire circle with math in it adds to what? Do the math please I'm wanting more specific than 63% what percent did math and English homework? See the and What's the and 16? What percent did math or English homework? now From the formula that means One plus the other minus the overlap, but from the quick Venn diagram I think it's that plus that plus that it's anything that appears in Those circles, so I'm just quickly going to go 47 plus 16 plus 25. What's the correct answer? 88 can you tell me another way I could have gotten the answer without adding 100% 1 as a decimal minus 12 percent point one two as a decimal Now if I had used the formula by the way to get or probability of write this down, please M or E it would be probability of E of M Plus the probability of E minus the overlap and it would have been Right does that match this equation up here? Yep, except instead of using A and B. I'm using M and E So M or E first one second one overlap and What did we say the first one was? 63 plus What percent did English homework not 25? 41 Minus the overlap does that also give you 88 check me does it also give you 88? Yes. No. Yes Yeah, which way is fastest compliment? Which way second fastest? But for what it's worth the formula would also have gotten me there if they'd given me that Two more with them a single card is drawn from a standard deck of 52 cards Use formulas to determine that the probability that a Nine of diamonds or a heart is drawn Do you know what word is that right there in the middle? So at the top of the page, let's write down the probability of a or B is equal to the probability of a Plus the probability of B Minus the overlap There's our generic template Instead of a Amy can I use? 9d for nine diamonds and Instead of B what would be a clever letter to use? H for heart Okay, so this is wanting to find the probability of 9d or H and that's going to be the probability of The first one plus the probability of the second one minus the probability of Both how many nine of diamonds are there in the deck? one out of How many cards are there in the deck? 52 52 52 Plus How many hearts are there in the deck? 13 divided by how many cards are there in the deck? 52 Minus How many cards are nine of diamonds and hearts at the same time? I think that's a trick question I think they're mutually exclusive Zero and This is why I said don't reduce fractions because technically this is one quarter, but if I had reduced this Don't they have a common denominator already built in if I don't reduce In fact, what is 1 over 52 plus 13 over 52 in your head with no calculator? 14 out of 52 good enough now let's compare that with B Instead of a nine of diamonds or a heart Just a nine or a heart Okay, that's going to be the probability of nine or H is going to be the probability of nine Plus the probability of H minus the probability of nine and H Right. I'm just filling in that template formula. I rarely use a and B I usually try and pick letters that makes or symbols that make sense to me. All right Evan What's the probability of a nine? four out of 52 plus 13 out of 52 minus How many nines are hearts at the same time one out of 52 Four plus 13 minus one 17 take away one 16 out of 52 So here's a question If you had to bet which of these two would be a better bet Yeah, B. They're both lousy by the way because they're not above 50 percent. They're both terrible bets So I guess I should really say instead of asking which of these would be the better bet Which of these would be the less terrible bet B? The odds are slightly better barely but slightly and I don't think you would see that intuitively just by glancing at it Example seven two hundred people with neurology symptoms Which includes headaches and backaches back aches participate in a study to evaluate a pain relief medicine 60 people experienced headache relief 126 experienced backache relief 36 experienced both How many categories are there in this question? There's two headache and backache This is where a Venn diagram not a formula shines I used a formula with the previous one because in cards there's all sorts of catty black red face Forget it here. What I'm going to do is I'm going to draw a Venn diagram like this with some overlap and whenever I'm doing a Venn diagram I want to try and find the overlap first if at all possible How many people are in the overlap category? How many people both? 36 I'll put a little 36 right there. How many people got relief from headaches? 60 now a lot. Oh, I'm going to call this one headaches and I'll call this one back aches in my high-tech numbering system And a lot of people Doug want to put a 60 right there. That's wrong Why is that wrong? How many people do you have in your headaches by putting a 60 right there? Not 60 How many do you have? 96 so I don't put a 60 right there What number is going to go right there? Gotta be 24 Can't be 60. Oh Kelvin what do those two together add up to? 60 there's your 60 that got headache relief 36 got both only 24 it got their headache, but it didn't get their back ache. Oh 126 people got backache relief. I'm not gonna put this is wrong. I'm not gonna put 126 there What's gonna go there? 90 oh Fatali how many people were in this study read the question how many people were in the study 200 So how many people did this drug not work for? Do the math? How will you calculate that? 200 minus 24 minus 36 minus 90 how many people did this drug not work for and This is some of the basic math behind clinical drug trials and things like that How many? anyone 50 5 0 or 1 5 5 0 Okay Now I've got my bent diagram set up now. I can answer the questions What's the probability that you get at least one of the two symptoms now at least one of means? headache or Backache because remember we said Or means one or the other or both now we could do this with a formula But isn't it gonna be? 26 plus 36 plus 90 or Better yet, isn't it gonna be 200 minus 50? I? think the probability That someone got relief from at least one symptom is that 75% I heard somebody say yeah, which is for a medication pretty good What are the odds that it didn't work for you now? How would I write this? I would write this as Not a or B. That's the abbreviation for neither and it's 50 out of 200 so we've thrown a lot at you terminology and Then diagrams and the or equation. I think you folks are getting the well Okay, I guess number one. It's gonna skip it, but no for skip five C So five a and b six seven eight nine ten Any election? Yeah, I'm gonna go 11 12 There's not gonna be as much homework next class. It's a lot I threw at you. I know but we're trying to Get the sprint going for the homestretch and you do have about Almost 15 12 well, yeah 15 almost minutes to work