 Today we're going to talk about probability and we're going to focus on and or and not statements So discussed to discuss and or and not statements. We're going to look at a local establishment That I'm a big fan of called sports cars are us so the other day I went to sports cars are us and Winston Salem and documented all the cars on the lot for whether they were bright red or Had been written a ticket Naturally, there were a few cars on the lot that were both red and had been written a ticket I found that there were nine red cars and that's going to be important in a minute There are 13 cars That had been written a ticket And then there were three cars that were both three cars that were both red and Had been issued a ticket Additionally, there were five cars On the lot that were neither red or or had they been written a ticket So to help process this data to help process this data I Constructed a Venn diagram, so I have the Venn diagram here So each dot on the Venn diagram represents a car this circle here Represents red cars and then this circle here Represents cars that had been ticketed so we can count and see that there in one two three four five six seven eight nine dots in the Red car circle, so there are nine red cars. We can count and see that there are one two three four five six seven eight nine ten eleven twelve 13 ticketed cars 13 cars in the ticketed circle and then that there are three cars in the intersection here Meaning that they are both red and they have been issued a ticket and then there are five cars Floating out here in the universe that are neither red Nor have they been ticketed So with all of this in mind Here's the question Let's say you go through a mini mid-life crisis You do not expect it the last long only a few days Because of your mini mid-life crisis you call up sports cars are us and ask them to deliver a sports car any sports car For the weekend to help you get over your mini mid-life crisis What is the probability that? a They deliver a sports car that is red and has been ticketed B they deliver a sports car that is red or has been ticketed and See they deliver a sports car that is red but has not been ticketed So we're going to start out by looking at Part a So part a they deliver a sports car that is red and has been ticketed So just as a refresher there are nine red cars 13 cars that have been written a ticket three cars that are both and Then five cars that are neither So what a lot of students start out doing if we wanted to if we want to determine the probability We need to figure out the total number of cars so the way The way we find the probability we look at the number of successes Over the total number of outcomes. We need to figure out the total number of outcomes We need to figure out the total number of cars So a lot of students start by saying oh i'll do nine plus 13 To get 22 and then i'll add the five For the cars that have not yet been counted. So that gives me 27 All right, we'll come back to this number in a second, but let's look again at our Venn diagram Let's see if i can pull up the Venn diagram here. Okay. I'm going to go through and actually count the number of cars here So there's 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 There are 24 cars So what was the issue here? Why did we get 27 when the actual number? is 24 And the reason for that Right now We have double counted the three cars that are both red and have been written ticket tickets Are being added here And here it's being added twice. So we need to subtract The three cars That uh Are both red and have been written tickets. So we don't double count So we end up with 24 total cars So What did I do? Let's see here's some space. So what did I do to come up with that number? I looked at the number of red cars plus the number of ticketed cars I subtracted The three cars that were both And then lastly I would need to add The five cars that are Neither Okay, and in this case I come up with 24 And then we can see that there are three cars here that are both. They are both red And have been ticketed So the probability of being Both being red and being ticketed would be three Over 24 And that would Reduce We could say 0.125 So the probability of randomly being Uh given a car that is both red and has been ticketed is 0.125 Moving along the part B Part B They deliver a sports car that is red Or has been ticketed So it's no longer an and statement. It's now an or statement a sports car that is red or has been ticketed We just determined there are 24 Total cars So how would you figure out the number that? Is are red or have been ticketed? and again you could Do nine plus 13 But once again you have to be careful about double counting. So you would then need to subtract three and get 19 so the probability of a car being red or ticketed would be 19 over 24 And if we were to look At the Venn diagram We would count one two three four five six seven eight nine 10 11 12 13 14 15 16 17 18 19 cars Uh a fallacy That a lot of students have here Is that they look at this they hear or And they want to exclude the cars That are both they want a car that's strictly red And a car or a car that's strictly been ticketed They want to exclude the cars that are both red and have been ticketed Um, that's not the case The example I always use Uh, I was a bit of a smart alec when I was Younger and I'd have a teacher say well, is this true or false? And I was always the one that would raise my hand and say yes, absolutely A statement is definitely true or false. It could be one of the others Sometimes we have situations where it's both and that is okay It doesn't mean uh, there's no sort of exclusive Exclusivity All right part c, uh, they deliver a sports car that is red But has not been ticketed So we we know there's still 24 cars How many are red but have not been Ticketed so if we take a look at the Venn diagram We're interested in red cars that have not been ticketed So we're interested in This portion here of the Venn diagram Now we're going to exclude these three cars That are both red and ticketed and we see that there are one two three four five six cars That are both red But have not been rickett written written tickets Flip back to black here and we see there's just six out of 24 Chants and That would come out to be about zero point two Five So you got about a one and four chance got a one and four chance. You got a point two five Chance of getting a car that is red but has not been ticketed