What's up my stat stars, welcome to the Unit 4 Summer Review video over Probability, Random Variables, and Probability Distributions. This is probably one of the toughest units that kids have in AP Statistics, so I'm going to do my best to make it the absolute best video so that you can review all the key things that you learned, and hopefully make it a lot easier for you. Now it's an absolutely enormous unit, and I've broken it down into three parts.
Part 1, Basic Probability Rules. Part 2, Discrete Random Variables. and their probability distributions, and part three, binomial and geometric probability distributions.
And this is only part one, so please stay tuned for part two and part three. Now, before we begin, let me remind you of two really important things. First, this is just a review video.
We're not going to cover every single tiny detail. In fact, probability is extremely difficult. with lots and lots of different intricate problems.
So this video can't cover every single scenario that you're gonna see. So please take a look at my YouTube channel for more videos over every single very detailed topic. The second thing I would remind you of is get out that study guide. If you haven't already, download, print that study guide, fill it out as you watch this video or fill it out when you're all done.
It's filled with lots of great practice problems. that correspond to what I talk about in this video. And not only this video, don't forget about part 2 and part 3 coming soon as well.
Alright, let's get ready to dive into it. Understanding probability begins with understanding what a random process is. A random process generates results that are random or simply unknown and determined by chance. Seeing patterns through collection of data through a random process doesn't necessarily mean that the variation we see is not random.
Random just means unknown and uncertain. An outcome is the result of a random process and an event is the collection of outcomes. Now when we say that an outcome is random, all we're saying is that we don't yet know what that outcome will be. For example, if the random process is rolling a fair six-sided die and we say that an outcome is random, all that means is we don't yet know what that outcome will be until we roll the die. Probability is simply trying to quantify the uncertainty in a random process.
One viewpoint of the probability of an outcome is the long-run relative frequency of that outcome occurring, meaning after a large number of repetitions of that random process, we could take the number of times we observe the outcome occur divided by the total number of repetitions ran. We can find these long-run relative frequencies, also known as probabilities, through simulation. Take rolling a die. If we want to find the probability of rolling a 2, we could start tossing a die repeatedly.
And after a large number of tosses, we could count the number of times we observed a 2 and divide it by the total number of times we tossed the die in total. But what is the long run? If we only rolled the die 20 times and counted the total number of times we observed a 2 divided by 20, we wouldn't get a very accurate probability. This is where the law of large numbers comes in. The law of large numbers states that simulated probabilities tend to get closer to the true probability the more trials we perform.
Basically, if we want to be more accurate, we need to run more trials. But how many is many? Well, a lot.
Infinite. That is why the long-run relative frequency definition of probability will never be perfect because we can never run infinite simulations or infinite trials. So just know that the more trials we run, the more accurate we will be to what the true probability could be. Now rolling a die is a really simple random process to repeat over and over. where other random processes are not so easy to actually simulate.
For example, let's just say the manager of a fast food restaurant knows that through past observations, 65% of customers who ask for a free cup for water take that free cup to the soda fountain and fill it with something other than water. I know, a crime. Now, maybe he wants to find the probability that of the next 10 customers who ask for a free cup of water, four or less of them fill it with something other than water.
Now, how would he actually simulate this random process? He would have to look at 10 customers, record how many actually filled their water cup up with something other than water, and then redo it again and do it again and do it again. And then he's going to count how many of those trials resulted in four or less people filling their free water cup up with something other than water. But that's really not feasible for the manager to actually do a large number of times.
So what could he do? while he could actually run a simulation using numbers. Running a simulation with numbers is where we pretend that numbers through a random number table are customers. And some numbers are going to designate that those customers are going to fill their water cup up with something other than water, while other numbers would represent that those customers are filling their water cup up with, well, water.
Then we could look at 10 customers and record how many of them fill up their water cup with something other than water, look at another 10 And because we're using numbers through a random number table or a random number generator, instead of actually looking at customers, we could do this really quickly and really fast. It actually generates some good probabilities. Then after many, many repetitions of looking at these 10 numbers, we could then count how many of our trials had four or less people fill the water cup up with something other than water, divide by our total number of trials, and we could get an estimated probability.
But again, we would need to run a large... number of trials to get close to the true probability of this event. Simulations can be very useful to try to find or at least begin to find what the true probability of an event is, but through some basic rules that I could teach you, we could calculate probabilities, true probabilities, really easily and really quickly. All we have to do is think.
To find the true probability of an event without Without a simulation, we need to first understand a couple basic ideas and some ground rules when it comes to probability. The sample space of a chance process is a list of all non-overlapping outcomes. For example, the sample space for the random process of rolling a fair six-sided die would be 1, 2, 3, 4, 5, 6. And we typically label observable events with Latin letters. For example, A equals we see a 5 on the next roll, or B equals we see a 2 on the next roll, or C equals we see an even number on the next roll. Now, finding the probability of event A, seeing a 5 on the next roll, is actually quite easy.
All we gotta do is take the total number of outcomes that are in favor of event A in the numerator, divided by, in the denominator, the total number of outcomes in the sample space. And for this situation, that'd be one outcome that's in favor of rolling a five, divided by six total outcomes in the sample space for a probability of one six. But you already knew that.
The probability of an event is always between 0 and 1, inclusive meaning it could be 0 or could be 1 or any number in between. Now, the complement of an event is that the event does not happen. And to find the probability of the complement, all you have to do is do 1 minus the probability that the event will happen. Makes it really simple.
So if the probability of an event A occurring is 15%, then you should automatically know very quickly without doing a lot of math that the probability that event A does not happen is 85%. Next, we have the probability of event A and event B occurring. This is looking for the probability that both events A and B occur at the same time, and we can express this probability in two ways.
This is also known as joint probability. Now, if the two events cannot happen at the same time, then we call them disjoint or mutually exclusive. And if two events are in fact mutually exclusive, then we automatically know that the probability of A and B is a big, beautiful zero.
Next, we have the probability of event A occurring or the probability of event B occurring, and we can express this using two different ways as well. This is known as the union of event A and event B. Now, when we're asked to find the probability of A or B, what we're really looking for is the probability that event A only happens.
or the probability that B only happens, or the probability that both A and B happen. Now, the complement of the probability of A or B is the probability that neither event happens. So if you think about two events, A or B, there's four things that can happen. A only happens, B only happens, A and B happens, or neither happen. And when we talk about the probability of the union or A or B happening, then we're looking at three of those things.
A happens, B happens, or both. That's why the opposite is neither. Now, finding the probability of event A and B occurring or the probability of event A or B occurring oftentimes drives students bonkers. Well, I can make it as easy as I possibly can, but first we have to talk about conditional probability. Conditional probability is when you are asked to find the probability of event A occurring given or on the condition that event B has already or will occur.
Now we express conditional probability like this. Before the line is what we're trying to find the probability of, and after the line is the event or the thing that we already know that has occurred or has been given to occur. And we have a really simple formula to find the probability of A on the condition of B.
All we're going to do is take the probability of A and B divided by the probability of B, the condition. So regardless of what letters we use, just know that the probability for conditional probability Or the formula for conditional probability is to take the probability of both events happening at the same time in the numerator, divided by the probability of the condition in the denominator. Okay, now we've gone through enough concepts of basic probability that we can talk about a couple really big probability rules.
The first major rule is called the addition rule, and it helps us find the probability of A or B. Here's how the formula for it works. To find the probability of A or B, we take the probability of A, plus the probability of B, and we subtract the probability of A and B. In simple terms, the word or means to add when it comes to probability.
But you have to consider that if the two events have some overlap, meaning they're not mutually exclusive, you have to subtract away the overlap, which is the probability of A and B. Now, for some, this can be hard to follow. Recall the probability of A or B is the probability of only A only B, as well as the probability of A and B. So if the two outcomes A and B have some overlap, again not mutually exclusive, that means inside of the probability of A is some overlap with the probability of B, and inside of the probability of B is some overlap with the probability of A.
Now we want this overlap to count, but we don't want the overlap to count twice. Seeing this through a Venn diagram makes it really easy to understand. If the events are mutually exclusive, there is no overlap. That is to say the probability of A and B equals zero. So we see two non-interlinking circles.
So the probability of A or B makes complete sense. It's simply the probability of A plus the probability of B. If the events are not mutually exclusive, there is overlap.
If we were to simply add the probability of A plus the probability of B, then the overlap that is in both gets counted twice. That is why in the addition rule, we subtract the probability of A and B away. We subtract the overlap, not to get rid of it, but to stop it from being double counted.
So when we subtract, we get the probability of only A, only B, and A and B. The second huge big rule is called the multiplication rule. The multiplication rule states to find the probability of A and B, you take the probability of A, times the probability of B given that A has already occurred. In simple terms, the word and in probability means to multiply, but you must always consider how the second event is impacted by the first event, which is why conditional probability is needed. Probabilities of multiple events are as simple as multiplying the probabilities together.
As long as you consider how the probabilities are affected, as the event progresses. This brings me to the final major topic of introductory probability, and that is independence between events. Two events, A and B, are independent if and only if, knowing that event A has occurred or will occur, does not change or impact the probability that event B will occur.
Basically, two events are independent if the probability of A equals the probability of A on the condition of B, and the probability of B equals the probability of B on the condition of A. If the probability of A is the same regardless of if event B occurs or not, then the events are independent. This brings me back to the multiplication rule. To find the probability of A and B, and you know that A and B are independent, then all you have to do is take the probability of A times the probability of B.
But that's only if they're independent because they're not affecting each other. But if you know that they are not independent or you're unsure, to find the probability of A and B, you must take the probability of A times the probability of B, given that A has already occurred. By now you may be pretty confused, but here's the one big recommendation I have.
You can never go wrong if you write out everything that can happen explicitly, and at that point usually probabilities are pretty easy to figure out. But To try to get through some of the confusion, let's take a look at a couple AP-style probability questions and make sense of them using our rules. All right, the first probability question we're going to take a look at deals with a two-way table. Now, the AP Statistics exam loves working with two-way tables and probability.
You'll see a couple of them scattered throughout the multiple choice, and you might even see one on the free response section. All right, here's the problem that we're going to take a look at. We surveyed 500 high school students, asking them two questions.
What high school do you go to? High school A, high school B, high school C. And what is your favorite flavor of slushy? red, blue, green, or neither. Now, with a two-way table, we could ask you so many different probability questions.
So, let's begin. The first question is, one student is selected at random from those surveyed. What is the probability that the student is from high school A? Now, we've got to remember the basic probability rule, right? In the numerator goes the number of outcomes in favor of what we're looking for, high school A, that's 140 students.
divided by the total number of students in the sample space. In that case, it's 500. 140 divided by 500 is 0.28 or 28%. So the probability that a student is picked and that student goes to high school A is 28%.
Second question is, again, a student is picked at random from those surveyed. What is the probability that the student prefers a red slushie? Well, once again, in our numerator is the number of kids that meet or are favored to what we're looking for. That's 275 kids we could clearly see in the two-way table that prefer a red slushie.
Out of the total sample space of 500 students, and we get 55% of students surveyed love a red slushie. The third question posed to us, once again, students selected at random from those surveyed, what is the probability that that student loves a red slushie and is from high school A? Once again, remember, we see that word and we're looking for what's called joint probability.
So we're just looking at where that row and column are joined together. So we see that there's a total of 91 students that are both in high school A and love and orange, or excuse me, orange, red slushy. And again, you don't have to do any sophisticated math to figure that out.
All you got to do is look where those two rows and columns are joined together. 91 out of 500 is 18.2%. Now, the fact that we have 18.2% of kids that Both, go to high school A and love a red slushie, means that the two events, go to high school A and loving a red slushie, are not mutually exclusive because we literally have 91 kids that do both. And our next question once again starts off with, given that a student is going to be picked at random from those surveys, this time it's what is the probability that that student is from high school A or likes a red slushie?
Now I can also say it the other way around, what is the probability that they like a red slushie or a high school A? or from high school a that doesn't matter the order there now remember in basic probability really simple way or means add but we got to make sure that if they're not mutually exclusive which we just said they weren't then we subtract away any overlap so to find the probability we're going to start off with the probability that a student loves a red slushy 275 out of 500 we're going to add the probability that the student comes from high school a 140 out of 500 but 91 kids just got double counted See, 91 kids were counted amongst the 275 that like a red slushie, and 91 kids were also counted amongst those that are from high school A. Those 91 kids that both are from high school A and love red slushies deserve to be counted, but they don't deserve to be double counted.
So that's why we subtract away the overlap. Again, we're not getting rid of them. We're just preventing them from being double counted.
So once we do the math, we already got a common denominator. It's pretty simple. 324 out of 500. or 64.8%.
Next question from this two-way table is what is the probability that the student prefers a red slushie, but there's a condition given that the student is from high school A. So I noticed I capitalized and I underlined and I bolded the word given, but again you got to make sure that you understand when you see that word given, you're working with the conditional probability. So we're trying to find the probability that the person likes a red slushie, but then we have our line for the given and then... after the line is the condition that they come from high school A. Now don't forget we have a pretty simple formula for conditional probability.
The numerator is the probability of both things A and B, in this case red, slushy, and from high school A, divided by the probability of the condition which they're from high school A. So following the math and the numerator is the probability that they are both from high school A and love red slushies. Once again, that's 91 kids out of 500 divided by the probability they are from high school A, which is 140 divided by 500. If you're really good at math and dividing with fractions, you'll see that those 500s are going to cancel. So we get 91 out of 140 or 65%. Now you actually don't even need the conditional probability for this formula or for this specific problem if you truly understand conditional probability.
So the idea is that we're given that the student is from high school A. That means we're only allowed to look at that row for high school A. We're not allowed to look at the total because that got kids from high school B and high school C. We obviously don't want to look at the rows for B and C because we're given that they must be from high school A. And that changes our denominator to 140. We're only allowed to look at those 140 kids from high school A.
And of those kids from high school A, 91 love a red slushie, so 91 out of 140. So you could logically think through this problem to get your answer, or you could certainly use the conditional probability formula and it still works to get your final answer 65%. Now the last type of question that the AP test loves to ask working with a two-way table is, is there independence between two events? So here we're going to stick with the two events, going to high school A and liking a red slushie.
And are these two events independent? Well, it's really simple because we have a formula that we could check to see if they're independent or not. And it's really simple. In generic terms, remember to check if two events, A and B, are independent.
We have to find the probability of A and compare it to the probability of A given B. If they're equal, then it just goes to show that B didn't matter at all. Hence, they're independent.
So here, what we're going to do is we're going to find the probability that the student loves a red slushie, which we've actually already dead. We already know that's 55%. 55%. 275 out of 500 kids love a red slushie.
No condition attached there whatsoever. Then we're going to compare that to the probability that the student loves a red slushie given that they're from high school A. So what we want to do here is we want to see does being from high school A change the probability of somebody loving a red slushie? If it changes it, then we have two events that are certainly not independent.
And well, we just got done actually finding that conditional probability. Again, remember the conditions that they're from high school A, so there's 140 total students from high school A, and of those 140 students, 91 love red slushies, 91 out of 140 is 65%. So what we see here is that there's a 55% probability of a student preferring a red slushie. However, if the student is from high school A, the probability they prefer a red slushie is 65%. So being from high school A...
makes a student more likely to prefer a red slushie, not independent. If one event changes the probability of another, then they clearly somehow depend on each other in their inner mix, so they cannot be independent. So in this situation, the two events, loving red slushies and being from high school A are certainly not independent. Now, obviously I focus on being from high school A and liking red slushies for this problem, but I could have worked on blue slushies, green slushies, high school B, high school C, or any combination of that to ask some really good probability questions from a two-way table. But I gave you the key ones that you're probably gonna see whether it be multiple choice or an FRQ.
All right, here we have a really good probability question. The probability that Andrea purchases car one is 42%. And the probability Andrea purchases car two is 26%. She's got two cars. She might purchase car one.
She might purchase car two. She might purchase both of them. She might purchase none of them, but we're given a couple of easy probabilities here. Now, assuming the decision to purchase each car is made independently, which means if she buys car one, has no impact on car two or vice versa, what is the probability she purchases exactly one of the two cars?
Now, what a lot of people do wrong here is they just jump in saying, oh, we're just going to add these two probabilities together. She wants to purchase one of the two cars. We got car one, we got car two, just add them. But no, that's actually not going to work here because remember, or probability, or probability means that, well, we want the probability of car one only, car two only, or both.
Well, based on what the question is asking, we don't want both. We just want exactly one. So remember my suggestions is to write everything out in the sample space and usually probability becomes pretty easy.
Easy. So what are some things that could happen? Well, she could buy car one, which we know is 42%, which means that the probability that she does not buy car two is 58%. That's just the compliment rule. The probability she buys car two is 26%.
So once again, the compliment rule says the probability she does not buy car two is 74%. Now what is the sample space here? Well, she could purchase car one.
and not car two. Again, walk through that. She's going to purchase car one.
42% and not purchase car two, which is 74%. Why am I allowed to multiply those? Because they're independent.
When events are independent, you can multiply all you want without too much thinking. So we get 0.3108. Another option is she purchases car one and car two. That's gonna be 42% for her to purchase car one times 26% for her to purchase car two.
We get 0.1092. Another option is she purchases car two and not. car one again that's gonna be 26 for purchasing car two and a 58 for not purchasing car one multiply those together we get 0.1508 and the last scenario is that she doesn't purchase either car 58 she doesn't purchase car one 74 she doesn't purchase car 2.4292 now let's go back to what the question asked the question says what's the probability she purchases exactly one car So of these four scenarios, two of them lead to her purchasing exactly one car. That would be car one and not car two, 0.3108.
Car two and not car one, 0.1508. Add those together because only one of them could happen, one or the other, and we get the total probability that she purchases at least, or not, excuse me, just exactly one car. The other two possibilities is she purchased both of them. Well, that's not what the question asks. Or she purchased neither of them.
Again, that's not what the question asks. So the final probability, once we add those two together, is about 0.46. All right, let's take a look at another problem with Andrea and her two cars, but there's a really big difference that matters so much. I really want you to take a look at this.
All right, the probability that Andrea purchases car one is 42%. So instantly we know the probability she does not purchase car one is 58%. The compliment, that's easy.
But this time, listen, if she purchases car one, the probability she purchases car two is 15%. The way that question is written is conditional, right? If she purchases car one, so we would need that to happen first, then the probability she purchases car two is 0.15. That's instantly telling me that what happens to car two is dependent on car one, so these are not independent like they were in the problem we just looked at. Now, the second thing it says is if she doesn't purchase car one, the probability she does purchase car two is 0.60.
Once again, what is the probability that she purchases exactly one of the cars? Cars. So again, let's break down what we were told. We know that there's a 42% chance she buys car one.
The complement means a probability of 58%. She does not buy car one. And we know a couple of conditional probabilities.
Given she purchases car one, the probability she purchases car two is 15%. Given she does not purchase car one, the probability she purchased car two is 60%. Now, the best thing we can do with a problem like this is make it really easy with a tree diagram to look at our four different scenarios. So in this tree diagram, we see first off the two original branches. She purchased this car one 42%.
She does not purchase car one 58%. Now, if she purchases car ones on the upper branch here, we could purchase car two or Andrea could purchase car two, which is 15%. That means that we also know the compliment of that, which would be there's an 85% chance she does not purchase car two, given that she does purchase car one.
Then in the bottom branch, we have her not purchasing car one 58%. And if she does not purchase car one, the probability she does purchase car two is 60%. And again, the complement of that would be 40% chance she does not purchase car two, given that she does not purchase car one.
Now to get to the end of any branch, all we got to do is multiply. And we are allowed to multiply these probabilities together because they're conditional. Again, you're always allowed to multiply when you're doing multiple things. She's purchasing multiple cars, so we want to multiply. Why?
As long as you think conditionally, which is how we create this tree diagram so that we could think conditionally. So again, what two branches lead to her purchasing exactly one car? Well, she could have car 1 and not car 2, so follow that branch, 0.42 times 0.85, 0.357. The other branch is not purchasing car 1, but purchasing car 2. Follow that branch, 0.58 for not purchasing car 1, 0.6 for purchasing car 2, 0.348.
eight. Now the top branch leads to purchasing both cars. That's not what they asked for.
The bottom branch leads to purchasing no cars. Again, not what the question asked. Now again, the question could ask that, but that's not what I asked you.
So two branches lead to exactly one car. All we got to do is add them together because it's one or the other, or it means add, they can't both happen at the same time. So we'll add those two probabilities together to get our final total answer.
And that would be a beautiful, all you got to do is add a point. 357 plus 0.348 to get 0.705 as our final probability for this particular question. So two questions that are very similar, but it just makes a huge difference if the events are independent or if the events are not.
Now we're going to take a look at a couple of what I call generic probability questions, because we just have generic events A and B with no real world problems or words attached to them. But the AP test loves to put these on the multiple choice questions just to make sure that you know how the different formulas work with each other. So let's take a look at a couple.
In this first generic problem, we are given two events A and B, and we're told a couple of things. First, the probability of A is 35%, and the probability of event B is 20%, and we're told that we need to find the probability of A. A or B, knowing that A and B are mutually exclusive.
Okay, first we're going to start off with the fact that they're asking us to find the probability of A or B, so we know we're going to need our OR formula, one of the most important formulas when it comes to probability. The probability of A or B equals the probability of A plus the probability of B minus the probability of A and B. Now next up comes another super important piece of information, that they are mutually exclusive. Mutually exclusive means that the two events have no outcomes in common and cannot happen at the same time, which instantly tells us the probability of A and B is a beautiful zero. So now we know everything we got to do to fill in the formula.
The probability of A is 0.35 plus the probability of B is 0.20 minus the probability of A and B, which we just said is zero because of the fact of mutually exclusive. And that makes the math really, really simple. We're just going to take 0.35 and add 0.20.
to get 0.55 or 55%. So the probability of A or B is 55%. And this next problem is a very similar setup.
We're given two events, A and B. The probability of A is 0.35. The probability of B is 0.20.
And we're asked to find the probability of A or B. But this time it tells us that A and B are independent. All right.
So where do we begin? Well, once again, it does ask us to find the probability of A or B. So we're going to need that. or formula.
The probability of A or B is the probability of A plus the probability of B minus the probability of A and B. So we can easily fill in the A with 0.35 and the B with 0.20. Now, what do we know is true when the probability of two events, or excuse me, when two events are independent? Well, when A and B are independent, it means that to find the probability of A and B, all you got to do is multiply the probability of A times the probability of B.
Now, normally, we got to think about, well, does A change B? But remember when they're independent it doesn't a and b don't affect each other and that is why when they're independent we know that the probability of a and b is automatically the probability of a times the probability of b so we could actually since we already know the probability of a and b we can easily find the probability of a and b all we gotta do is take 0.35 multiply it by 0.20 to get 0.07 so now the probability of a or b is 0.35 for the probability of a plus 0.20 for the probability of b minus 0.07 for the probability of A and B. Put all that together, we get a final answer of 0.48 or 48 point, or excuse me, 48%. Now this next problem looks really similar, but it's a little bit different. And because it's a little bit different, it's gonna make us have to do some dreaded algebra.
But thank goodness the algebra doesn't get too difficult. So once again, we're given two events, A and B, and we're told the probability of A is 0.35 and the probability of A or B is 0.80. And we're asked to find the probability of B. And in this first problem, we're told that A and B are mutually exclusive.
All right, so this time we know the probability of A or B. We know the probability of A, and we're asked to find the probability of B. So since we know the probability of A or B, we're still going to want to use that OR formula, the probability of A or B equals the probability of A plus the probability of B minus the probability of A and B.
Now... The other piece of information is super important is that they're mutually exclusive, which automatically tells us that the probability of A and B is a beautiful zero. So now we just got to fill everything in.
This time we already know the probability of A or B is 0.80, equals the probability of A, 0.35, plus the probability of B, which we don't know, so we can either turn it into an X, which you would do in algebra, or you could just leave it as a probability of B, minus zero, because the probability of A and B is zero when you're mutually exclusive, Now we've got a really simple algebra equation to solve. All we've got to do is subtract the 0.35 to the other side to get 0.45. So the probability of B is 0.45. And our final generic question looks like this.
Given two events, A and B, the probability of A is 0.35. The probability of A or B is 0.80. We're asked to find the probability of B, but this time A and B are independent. All right, this one involves a little bit more algebra, so hopefully you can stay with me here. So first, once again, we know the probability of A or B, but we're still going to use that formula, the probability of A plus the probability of B minus the probability of A and B.
Now, we already know the probability of A or B, so we can substitute that in as 0.80. We already know the probability of A, so we can substitute that in as 0.35. And we're asked to find the probability of B, so we can either just leave that as the probability of B, or we can change it to an X if we really want to make this look like algebra. Now, what does it mean when the two events are independent? Well, once again, when two events are independent, it means that the probability of A and B is simply the probability of A times the probability of B.
We don't have to think about any kind of conditional probabilities. So we know the probability of A is 0.35, and we're going to multiply that by the probability of B, which we don't know. So we end up with this equation, 0.80 equals 0.35 plus the probability of B minus 0.35 times the probability of B.
Now what we have to recognize here is, again, algebra, we have some like terms. We have 1 probability of B minus 0.35 probability of B. Those are like terms. And if that's a little bit confusing to you, just turn them into Xs.
Make the probability of B an X. So we have one X minus 0.35 times X. Because again, probability of B is an X. So once again, those are like terms that we can simply combine.
One probability of B minus 0.35 probability of B is 0.65 probability of B. Just combining like terms with algebra here. It doesn't get too difficult.
Then we have, again, a pretty basic algebra equation to solve. We're gonna subtract the 0.55 from the 0.80 to get 0.45. Then to solve for the probability of B, we have to undo that multiplication with divisions.
We're going to divide 0.65 to the other side, and we get a final answer of 0.692 for the probability of B. So a really good problem. They love putting ones like this on the AP test because it forces you to do a little bit of algebra with some good probability work. Now, here's one final probability question I want to ask you because almost every single year in the AP stats exam, there is a problem.
like this of some shape or form, whether it be on a multiple choice or FRQ, but it's one that challenges students, but if you know how to do it, it's actually pretty easy. Here's the problem. 2% of a population of men have a disease.
There is a test that men could take for this disease, but no test in the medical field is perfect. So if a man has the disease, the test will give a positive result saying they have the disease 95% of the time. If the man does not have the disease, the test will have a positive result 12% of the time. Well, that's called a false positive because it says it's positive, but remember, the man doesn't have the disease.
What is the probability a man has a positive result? Now, what a lot of students want to do right away is they just want to say, oh, 95 is positive, 12% positive, add those together, but then you get a number bigger than 100%, which in probability world, that's impossible. So let's take a look at this problem.
Let's really think about all the different probabilities given. First, 2% probability that a man has the disease. Well, the compliment would be the man does not have the disease, 98%. That's going to help us. Now, if the man has the disease, 95% of the time, it'll have a positive result.
That's a true positive, a positive that is accurate. But the compliment of that would be the 5% chance that it comes back negative. That would be a false negative.
It says you don't have the disease, but remember, you actually do. Then we have that 12% positive, but that's if you do not have the disease. That means a... true negative would be the 88%. That's the compliment of the 12%.
88% would be the probability that you have a negative result when again, you do not have the disease. Now let's walk through the different outcomes that are going to be what we want, a positive result. And that's going to get us the probability we're looking for.
So first a man can have the disease 2% and the test comes back positive 95%. So just follow that through 0.02, have the disease. and then there is a positive result, 0.95, and we get 0.019. Once again, why are we multiplying? Because I'm using the word and.
And means multiply. And you can always multiply when you see the word and as long as you think conditionally. That 95% is on the condition of the 2% happening first. That's why I had to write it that way.
Now, the other scenario that leads to a positive result is you do not have the disease, 98%, and you still got that 12% chance of a positive result when you. do not have the disease. So once again, 0.98 times 0.12, 0.1176. Now only one of these situations can happen. They both can't happen.
They can't both happen at the same time. That'd be impossible. So it's one or the other.
And remember that word or means add. So we're going to add those two final probabilities together to get a final answer for the probability of a positive result, 0.1366. Now, here is another very common question in a situation like this. Given that a man gets a positive result, so a man goes to the doctor and says, hey, I want to see if I have this disease or not. Doctor gives him a test.
So given that the man gets a positive result, test comes back positive, what is the probability he actually has the disease? Now, a lot of kids who want a multiple choice are going to guess 2% because they go back to the problem. The problem said 2% chance they have the disease, and they're just going to answer that.
But yes, there is a 2% chance a man has a disease, but this is conditional. It says given that they have a positive result. So we want to find the probability that the man has the disease given that they get a positive result.
So we are going to need our conditional probability formula, which is really easy and it's on your formula sheet. In the numerator, we have the probability of both the disease and getting a positive result. In the denominator, we have the probability of simply the condition having a positive result. Numerator is really easy. So have the disease and positive.
Just follow that scenario. Have the disease 0.02 and 0.95. Easy. Multiply and we get our 0.019, the numerator.
Now the denominator is the probability of getting a positive result, which we just got done figuring out 0.1366. Let me walk you through it one more time. Have the disease 0.02 and test positive 0.95. Multiply or, because they can't both have the same time, so we're going to add, not have the disease 0.98. times still getting a positive result, 0.12.
Multiply, add, we get that 0.1366 in the denominator. Then we just have to divide everything, divide 0.019 by 0.1366, and we get our final answer, 0.1391. So if a man gets a positive result, don't worry too much.
You only have about a 14% chance of actually having the disease. That's still kind of high, to be honest, but it's not like a positive result instantly means you're gonna have the disease because so few people actually have the disease in the first place. All right, that's it for part one, basic probability. Hopefully those examples made sense to you.
Now, if you're like, really, seriously, you're only gonna do four or five examples? Well, yeah, this is just a review video. I don't have time to go over every single type of probability question that you're ever gonna see.
But guess what? Part two is over discrete random variables and probability distributions. So we're still going to do a lot of probability in part two of the unit four summary review video.
And then don't forget about part three over the geometric and binomial distributions, because guess what? Those two different distributions still involve a ton of probability as well. So keep watching part two and part three, and you'll still get a ton more practice with probability.
I'll see you in part two.