 So today lesson three we're going to talk about the concept of or and what I mean by that is we're going to ask ourselves What if the question says what are the odds of this or that happening? And as a subheading you may see under objectives It says when do you add probabilities? And if you want a shorthand way to remember that or means add and I've kind of threw that in last unit already with combinatorics I said that and means multiply last you know, which is really going to become important this unit Jesse and or Means add but it's not White that simple So the question asks when do you add probabilities? And we have here a lovely generic then diagram If each of the 13 outcomes in the sample space is equally likely find the following probabilities by counting the outcomes So what's the probability of event a I said last day if you can count it you can solve it What's the probability of whatever the heck event a is? What is the probability? four out of 13 What's the probability of event B? three out of 13 What's the probability of a or B? seven out of 13 What's the probability of a and B? What's the overlap? Zero and then over here. We have a key term that you're going to need to know you underline it. I'm gonna highlight it mutually exclusive We say that looking at this then diagram we say that events a and b are mutually exclusive because There is no Overlap that's from the Venn diagram what we're really saying is or the probability of a and B is Zero they can't both happen at the same time if event a is a red card an event B is a club What's the probability of getting a red card and a club at the same time from one card? Zero you can't be a club and be read at the same time you can be a club You can't be read okay, if event a is Rolling a one and event B is rolling an even number What's the probability of rolling a one and an even number at the same time? You get the no overlap you can't Then it asks What's the relationship between the probability of a or B the probability of a and the probability of B? How are these numbers related? Well, I'm noticing the probability of a or B is really the first one plus The second one or as I hinted last unit or means odd Because four out of 13 plus three out of 13 is seven out of 13 Situation two so we have another lovely Venn diagram What's the probability of a here don't say three out of 13 because it's not It's still one two three. There's still four dots in this circle What's the probability of event B? five Out of 13 what's the probability of event a or B? How many dots are in one or the other or both? eight Out of 13 What's the probability of a and B? What's the overlap? one Out of 13 Can you see any kind of relationship between these four numbers? Specifically if I said I'd like to come up with an equation the probability of a or B is equal to what? probability of a Plus probability of B That gives me nine out of 13. I want an answer of eight out of 13 Oh minus the overlap and and this is the or equation This one here is actually the same equation There is a minus the probability of a and B right there But the reason it's invisible is what is the probability of a and B for this previous example? Zero and we don't write zeros if we don't need to this is the general equation This is the if you're mutually exclusive equation So events a and B are not mutually exclusive. Why are these not mutually exclusive? How can you tell this by glancing at the over at the over whoops at the event diagram? Okay, because they overlap or Probability of a and B is not equal to zero red card and a jack Can you be a red card and a jack at the same time? Yes, if you wanted to find the probability of getting let's see I did so well with the club last time. I'll go have it right there. I Got a red card 50 chance not that impressive actually But the probability of getting a red card or a jack is not a 50-50 chance. It's a little different or is it? so the addition law The general case the probability of a or B is the probability of a plus the probability of B minus The overlap so I fibbed a little bit last unit when I said or means add What I should have said was or means add minus any overlapping stuff that you counted twice oh and if they're mutually exclusive then Or does mean add and that's what we were doing last unit last unit anything that we were counting have no overlap specifically When we're doing committee questions, you weren't picking somebody twice for the same committee So they couldn't be the first one and the second one example one One card is drawn randomly from a deck of 52 cards event s is the card is a spade I'm gonna say I'm gonna do this in black And I'm gonna circle all of the spades. That's this one right here Event R is that the card is red. I'll do that in red all the red cards Are events s and r mutually exclusive? Yes, now it's easy to see from the picture You will have to just be able to visualize and use your imagination and most of the time It's common sense can it happen at the same time yes or no Event f is that we are a face card. I'll use blue now the face cards are this group here So it asks which of these three events are mutually exclusive. I think only two of them are I think s and r so it says Determine the following probabilities by counting the outcomes. What's the probability of a spade? I don't want it reduced, please 13 out of 52 because later on Tyler when you're using the addition law Jesse you want to help me out to your right? strong elbow he needs it. Oh He needs it. He was gone Tyler I don't want to reduce because if I'm using the addition law it won't I want it out of 52 with a common denominator Anyways, so that that's why I said to you guys don't bother reducing your fractions Do it at the end on your calculator if you need to find but meanwhile Oh, what's the probability of a red card 26 out of 52? What's the probability that we're a spade and a red card? What's the overlap? zero What's the probability that we're a spade or a red card? I? Think correct me if I'm wrong 39 out of 52 it's 13 times 3 probability of a spade we said was 13 out of 52 probability of a face card. Oh, what was the probability of a face card? How many face cards are circled? 12 out of 52 What's the probability that you're spade and a face card out of 52? What's the probability that you're a spade or a face card? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 I think Now that's doing this by Counting So we want to break away from having to count so I'll scroll up so I can't see the deck of cards Says use the addition law the probability of s or r is going to be the probability of s Plus the probability of r minus the overlap and oh by the way This formula here this one Sorry, this one the general case is on your formula sheet Although I kind of remember or means add minus overlap, which is going to be 13 out of 52 plus 26 out of 52 minus. Oh, we said the overlap was 0 and You can get the 39 out of 52 without physically counting s or f Gonna be the probability of s plus the probability of f minus the overlap So here's how I would do this question if I didn't have these numbers in front of me I would say to myself self how many spades are there in the deck? 13 out of 52 plus how many face cards are there in the deck? 12 out of 52 minus how many face cards are also spades? three Jack Queen King spades and this is where you also get your answer of 22 out of 52. So before we turn the page You got a little bit of room at the bottom here find the probability of red or Try this one on your own. It's going to be the probability of r Plus the probability of j minus the probability of r and and J 26 out of 52 Plus 4 out of 52 minus how many red Jacks are oh? 2 out of 52 30 minus to 28 out of 52 That's Tyler where the don't reduce your fractions really comes in here Yes 28 right Okay, are we gonna play any games? Oh just you wait One year back when I had an honors class and I was way ahead of the game when we were on semester We did a one-day casino, but unfortunately I don't think I'll be able to fit that this time and then what I did is every I think I had a little time or every five minutes when the timer went off They had to stop and calculate the odds of them winning that particular poker hand Texas hold them though which was not popular at that point much more difficult because then you anyways seven It would be a tree with seven level that there are 30 students in a class 16 turn the page 16 students surf the internet Ten students I really had to tell you to turn the page because there was a long delay there Okay, and ten. I know Dylan you like you make it a lot ten students use email Six do both What's the probability of a randomly selected student surfs or uses email? Okay, so if I read this correctly What's the probability that we're using the internet according to this question? What out of what? 16 out of 30 What's the probability that we use email? ten Out of 30 What's the probability did they give me both? Because and means both. Oh, yes, they did so the probability of I and E is six out of 30 What's this question want me to find the probability that they surf the internet or use the email so I know that probability of I or E it's going to be the probability of I Plus the probability of E Minus the probability minus. Mr. Do if not equals Minus the probability of I and 16 plus 10 take away six 16 plus 10 take away six is 20 That's the formula approach Another approach is to use a Venn diagram Now a Venn diagram takes a bit longer to set up I use a Venn diagram if the question has a part a part B part C because I can answer everything Instantly from the Venn diagram without having to redo the formula after the formula after what would the Venn diagram look like? I would draw this and I would say we have Internet email And if I'm using a Venn diagram, I either want to start from the very center and go outwards or I want to start from the outside and go inwards Okay, how many am I gonna put here? What's the both? Six how many kids surf the internet 16 does that mean I'm gonna put a 16 right here? No, what am I gonna put there Victoria 10 because that gives me 16 overall in this circle. What am I gonna put right here? For how many use neither what's going out here 10 Clearly this was typed at the beginning of the century. These were typed up in 2001 And you see now I can answer all sorts of questions. How many only use email, but don't actually surf the web What's the probability four out of 30 see it? What's the probability that they surf the web, but they don't have an email address 10 out of 30? So I use a Venn diagram There's 10 using email Right there Yep, and I'm glad you did that. Yeah, you need to realize These numbers won't blatantly appear because you always have to take into account the overlap Example three. Are they just asking one question or are they asking more than one question? I'm gonna use a Venn diagram here says what Wilma the web whiz submits bids on two web design projects She thinks she has a 70% chance of getting the first project But just a 50% chance of getting the second She puts only a 25% chance on getting neither So I'll call this the first project I'll call this the second project And I said to you I always like to start in the middle. I Always want to see if they have told me both Have they told me both? Jesse, you're correct. Say it loud So you know what I'm gonna do. You know what the odds of getting both is why it's X. I'll put something there Victoria in the previous question when we had a six there, how did you get this 10? you went 16 Take away six what number is gonna go here? 70 area gotta be oh and what number is gonna go here 50 minus X? Gotta be oh what hurt what's their chance of getting neither? 25 now they'll either give you the overlap or they'll give you the outside they have to give you one of those Okay, then I will here's our final little conclusion If I add this and this and this and this all together What percent should I end up with grand total at the very very end if I add up every single possible outcome? Gotta be a hundred percent. Let me say that again, but with math if I add That and the middle overlap and circle two and the 25 that has to add to 100 or In the previous example it would add to 30 if you're actually putting the people in there here It's percentages and usually my rule of thumb is if it's percentages I almost always fall back out of then diagram because I can also do most of the arithmetic just even my head I can subtract from a hundred or add to a pretty good I Have minus X plus X minus X when I go minus X plus X minus X two of my X's Cancel and I end up with just a single single solitary X oh, and I have 70 plus 50 plus 25. What is 70 plus 50 plus 25? Oh, sorry a minus X. Did I say a plus X? I'm sorry a minus X plus a hundred and forty five equals 100 or Thinking about it in a hundred and forty-five take away the middle equals a hundred What does X have to be? What's the middle got to be? Neither so now I'm gonna quickly very quickly Redo my then diagram. I'm not gonna redraw it. I'm gonna go cross that out. What's going there? 45 Cross this out. What's 70 minus 45? 25 Cross this out. What's 50 minus 25 45? Sorry 5 I can now answer every question Ready What's the probability that she gets both projects? 45% What's the probability that she gets at least one now at least one means one or the other or both? What's the probability she gets only the first one, but not both see it what? 25 what's the probability that she gets only one project, but not both? 30 no, that's both Says only the first project. This here is the first project and the second project Only the first means you get the first one, but you're not allowed to get both Yeah, that's at least one or that's at least one or that's at least at those three Those are all at least one of the two projects Yep, and you might have noticed Miguel as a shortcut I could have said well if the odds of her getting none is 25 percent the odds of her getting at least one is 75 percent It's the compliment because I'm that too Let's try a few more from the workbook If you are would be so kind as to turn to page 431 431 these two events right here mutually exclusive Blah blah blah blah blah blah blah 432 There's some with overlap What you want to look at is at the bottom of page 432 This is the formula that appears on your formula sheet. Oh And if there is no overlap what is and zero which becomes or modifies Jesse into that equation there I really don't even bother memorizing this I always just write this out and then oh and a zero great I cross it out Why memorize two different equations when one really does the trick and let's highlight the word mutually exclusive in our notes as well So when you're studying, you know what that was So very quickly we're not going to write stuff down. We're just gonna do these orally If we draw a card from a standard deck our events a and b mutually exclusive. Yes or no No, you can be a face card and a club So they're not there would be an overlap. I'd have to think that through if I was calculating this out To dice are thrown our Events a and b mutually exclusive the dice show the same value or what we used to call doubles and The total score is 11. Can you be doubles and get a score of 11? No That what does it have to be if you're getting a score of 11? What must your dice show? 65 right somebody say seven and four really? Oh boy Here are four events a b c and d which ones are mutually exclusive which ones have no overlap Which ones can't happen at the same time B and D. I agree another one a and c can't be a club and a face card Sorry can't be a face card and an ace ace doesn't have a face. There is no face on ace There I think I've been through that one before as my okay so Example four is the one I really want to look at Use the following information to determine whether they're mutually exclusive if they're mutually exclusive that means And has to be zero So if they're mutually exclusive they've given me the probability of a I don't know what it is probably of B I have no idea what event B is and they gave me or If and equals zero What we're really asking is I'll put an equal sign in with a question mark above it Does seven over 12? equal One quarter plus one third if it does Then and must have been zero and they're mutually exclusive And I'm not going to bother wasting my time with a common denominator One quarter plus one third equals Math enter enter. Oh, is one quarter plus one third seven over twelve it is So I guess I don't need the and oh that must mean that the and must have been zero There was no overlap Therefore mutually Exclusive then diagrams make things way easier What's the probability that a student did math homework from this Venn diagram? Don't say 47% because it's not Yeah, it's those two right those are both in the big math circle 47 plus 16 is 50 63 What's the probability that they did math and English homework at the same evening? 16% that's the overlap What's the probability of math or English homework that or that or that or means add add them up? Oh? You know what? Yeah, did you go 88 or did you go 100 minus 12? Probably faster to go 100% minus 12% everything's got to be in the middle 88% example seven okay This is an example of a kind of question that I would choose to use a Venn diagram not a formula Why I notice is a part a and a part B. I read the question really carefully Did they give me the overlap read carefully did they? What word tells you and what's another word for and begins letter B? Okay, both another word for and so I'm gonna do a little Venn diagram right over here And it looks like we have headache relief and Backache relief Except I did a really bad diagram. Let's make it a little bit larger mr. Do it so that I can There and I said whenever possible we want to start with both if not we put an X there and work our way out by subtracting what's both 36 So what's gonna go right here not 60 how many only cured their headache, but didn't cure their backache 24 how many got backache relief, but still had a headache 90 Now this is not percentages. This won't add to what this won't add to 100. What will this add to? The total number of people how many people did the drug not work for? Yeah, more specific 50 no 60 let's see 60 plus 90 is a hundred and fifty. I think the answer is 50 so Now they want the probability so the probability for part a is gonna be out of 200 How many got relief from at least one symptom? 150 how many got relief from neither or Or Formula pretty good then diagram actually handy or more It takes more time to draw it the formulas quick Which is why if they're only asking one thing usually I fall back on the formula But to ask more than one it's worth throwing the Venn diagram because everything falls out there homework Number one number one is just asking which of these are mutually exclusive. Yes or no It it's here, right? Think about it Four five is good Six is good seven eight Nine skip ten, but I have to do the election question since that's tonight 11 and 12 is good The answer is D for 11 Why and if you know what as I've said if you can do the homework in your head check the answer if it's right Don't bother writing it circle it and don't don't do any work because why waste time practicing what you already know We actually on Friday after school several of us teachers We had a duck together and tried to discuss What we thought homework would look like in the next 10 years and where we thought it was going because we really think in particular in elementary schools Like we have eight-year-olds coming home with an hour and a half of homework really I Don't remember that night anyways We tried to discuss what good homework is and what bad homework is and I know hopefully you guys have noticed I'll never give you something to shut you up or to keep you busy I will always assign what I think is what you need to know to master it But you need to adjust the homework to yourself