Probability is something that we deal with every single day without realizing it. Whenever we open our weather app to check for the chance of rain or when we consider the possibility of winning the lottery, we're really dealing with probability. But probability isn't just something that we can think about or consider. It's something that we can actually calculate. Now, I know this might not be the best news for you knowing that you have another thing to calculate, but you don't have to worry because here I'm going to walk you through the basics of calculating probability and some of the notation that goes along with that. So let's go ahead and get started. Now, when working with probability, mathematically, we consider the probability of different events and we rate this as P with the event in parenthesis. Now, when I say event, I don't just mean some big event like a party, an event is anything no matter how big or small that could happen. So if I was considering the probability of it raining, I would rate P with rain in parentheses. Or if I was considering the probability of getting heads when flipping a coin I would rate P with heads in parentheses. Now, when actually calculating probability, we need to consider the two different types. So if I have a coin in my hand and I'm going to flip it, but I want to know the probability of getting heads. If I do, since this is based on what could possibly happen, this is referred to as theoretical probability. But if I were instead to take that coin and flip it three times and record the results of each flip and calculate the probability of getting heads based on that. Since this is based on what actually did happen, this is referred to as empirical or experimental probability since we performed an here. Now when calculating the theoretical probability of getting heads, since I know there are only two possibilities when I flip a coin heads or tails and only one of those is heads. The theoretical probability of getting heads on a coin flip is equal to one half. Now, with our empirical probability over here, since we got heads on two out of those three tosses, the empirical probability of getting heads on a coin flip is equal to two thirds here. Now, the most important thing to consider here is that our theoretical probability was calculated above for any events actually happened. Whereas our empirical probability was calculated after our events actually occurred based on data. Now, when looking at our formulas that we actually used to calculate these here, these look slightly different just because they're tailored to each different type of probability, but they're really the same thing. So with our theoretical probability here, we took the number of outcomes that included our event. So in this case, there was only one way to get heads. And then we divided that by the number of total possible outcomes, their heads or tails. So we took our event and divided it by our total. But with our empirical probability, we took the number of times our event actually occurred in this case twice and divided that by the number of total trials. So we still took our event and divided it by our total. Now that we've seen the basics of calculating probability, let's work through some examples together. So looking at our first example here, we're asked when rolling a six sided die, what is the probability of rolling a number greater than three? So here we're looking for the probability of getting a number greater than three. Now when rolling a six sided die, I know that I could roll any number one through six. And when looking at these possible outcomes here, I see that only three of these numbers are greater than three. So when calculating the probability here, I would take the possible outcomes here that include my event in this case, three of them and divide it by the number of total possible outcomes in this case, since there are six sides of the there is six total possible outcomes. Now simplifying this fraction gives me a probability of one half and you'll often see probabilities reported as decimals. So here, the probability of rolling a number greater than three is equal to 0.5. Let's look at our other example here, here, we're given data and asked to find the probability of rolling a number greater than three based on that data of rolling a six sided diet 10 times. So here we're calculating empirical probability because we're actually given data to work with of events that actually already happened. So here we're calculating still the probability of rolling a number greater than three. But we're gonna take a look at this data table. So since there were 10 total rolls here, I already know that the number on the bottom of that fraction is going to be 10 for the total number of trials. Then I wanna look for the number of times that my event actually occurred since my event was rolling a number greater than three. I wanna look for the number of times that that happened. So on that first roll, I got a six, then I got a four and then I got a bunch of fives in a row and a six and then a four and a five. So eight out of those 10 times I roll the number greater than three. So taking that probability and simplifying that fraction, I get a probability of 4/5 or as a decimal 0.8. Now, something that you might be wondering here is why our empirical probability is so different than our theoretical probability when we're technically calculating the probability of the same event, and the answer comes down to our sample size. So since we only did 10 trials, this is not quite so close to our theoretical probability. But if I were to, to roll the dice 100 or 1000 times, I would get closer and closer to that theoretical probability of 0.5. So this is something to consider in your science courses when you're working with your own data. Now, when you're working through probability problems, you may see all of the possible events, possible outcomes of it of an event expressed as a set. So in curly brackets from set notation, now this will be referred to as a sample space and it just shows all of the possible outcomes in between those curly brackets. So for example, the sample space of flipping a coin, I would say that S equals S is the sample space. And then I would put all the possible outcomes in between those curly brackets. In this case, either heads or tails. So this here represents the sample space of flipping a coin. Now that we've seen the basics of calculating probability and we're more familiar with some of that notation. Let's get some more practice. Thanks for watching and I'll see you in the next one.

2

Problem

Problem

Given the data below, determine the probability that a person randomly selected from Group 1 will be wearing jeans.

A

0.37

B

0.48

C

0.52

D

0.63

3

Problem

Problem

In your coin purse, you have 3 quarters, 4 nickels, & 2 dimes. If you pick a coin at random, what is the probability that it will be a quarter?

A

0.33

B

0.44

C

0.50

D

0.66

4

example

Introduction to Probability Example

Video duration:

3m

Play a video:

Hey, everyone in this problem, we're told that whenever playing a particular lottery game, you must choose five numbers between one and 40 you win, if those five numbers match those drawn in the lottery. So we want to find the probability that you will win if you purchase one lottery ticket and if you purchase 50 different ones. So let's go ahead and get started here. Now, in calculating probability, we have to take the number of outcomes that include our event and divide that by the number of total possible outcomes. But here there are a lot of possible outcomes because there are a lot of different ways that you could choose five numbers between one and 40. Now, I don't want to have to write all of those out and figure them out manually. So we actually can use a tool that we've used in the past year. And that's our combinations formula. We can use this to calculate the all of the possible combination of five numbers between one and 40. So before we can calculate our probability, we first need to come up with a number of total possible outcomes using that combinations formula So here we see that our value for N is going to be equal to 40 because we're choosing out of 40 different numbers. But out of those 40 numbers, we're only choosing five. So that represents our value for R. So here I'm going to calculate 40 C five or the combinations of five objects out of 40 total possible objects. So here plugging these values into my formula, we're gonna go ahead and take 40 factorial and divide it by 40 minus five factorial times five factorial. Now, from here, we can simplify this to 40 factorial divided by 35 factorial times five factorial. Now, from here, if you know how to use the factorial button on your calculator, you can go ahead and just type this in and go from there. But here I'm going to walk you through how to do it by hand just in case you want to see that. So from here, we're going to rewrite that numerator in order to cancel out the highest factorial in our denominator, which in this case is 35 factorial. So doing that, we can rewrite that numerator as 40 times 39 times 38 times 37 times 36 finally times 35 factorial. Now all of this, of course is divided by 35 factorial times five factorial and that 35 factorial on the top and the bottom cancels just as we wanted it to. Now in my numerator, I'm left with this 40 times 39 times, 38 times, 37 times 36. And in my denominator, I'm left with five factorial. Now that five factorial can get expanded out into five times, four times, three times, two times one. So in my numerator, I have some multiplication happening as well in my denominator. Now doing that multiplication and then dividing that numerator by and by denominator, I get a final answer of 658,008 total possible combinations here. So this tells me the number of total possible outcomes when calculating my probability. Now we can go ahead and calculate our probability. So the probability of winning with one single lottery ticket, if I purchase one lottery ticket, there's only one possible way that I could win if my one lottery ticket matches what's drawn. So I'm going to take that one and divide it by my total 658,008. Now, this is a fraction might not mean much to you. But if we put this as a decimal, this gives me a value of 0.00000152. And this is our probability of winning with one single lottery ticket. So our chances don't look too great here. But let's look if we were to buy 50 different lottery tickets. So here, now the probability of winning there are now 50 different ways I could win if any one of my 50 purchase lottery tickets matches that, that matches that, that was drawn. So if I take that 50 still divide it by that same total 658,008, I get a decimal value of 0.000076. So a little bit higher possibility probability of winning the lottery here, but maybe not quite as high as we'd want to see. So now that we know how to find probabilities with also using our combinations formula. Thanks for watching and I'll see you in the next one.

5

concept

Complementary Events

Video duration:

4m

Play a video:

Hey, everyone, we just learned how to find the probability that some event will happen, like say rolling a six sided die and getting a four. But what about the probability that some event will not happen? Well, this is something that you'll actually be asked explicitly to calculate. And it may sound like it's going to be tricky, but you can actually do it using something that we already know. So here I'm going to walk you through how to calculate the probability that some event will not happen by simply using the probability that it will. So let's go ahead and get started here. Now all the possible outcomes where some event does not happen actually has its own special name and it's referred to as the complement of that event. So when looking at my dice roll here, if I consider all of the possible outcomes of not rolling A four, like rolling a 1235 or six, all of these outcomes together represent the complement of rolling A four. Now, if we refer to our event as a, we can use a special notation to denote the compliment of A. So you may see this written as a, with a little apostrophe after it or with a line over it or with a little symbol in front of it, that just means not all of these are different ways to denote the complement of a. Now that we know what the compliment is. Let's dive deeper into our dice roll example here. So in this example, we're asked when rolling a six sided die, what is the probability that we will roll a four? Now if we, we refer to this event as a, the probability of A is equal to the number of outcomes that include that event. So in this case, there's only one way I could roll a four divided by the number of total possible outcomes. So since this is a six sided die, all of my total outcomes are six. So the probability of A is 1/6. But what about the probability that we will not roll a four or the probability of the compliment of a? Well, I know that I have five possible outcomes that I would not roll a four because I could roll a 123, five or six. So here I take all of the total, all of the outcomes that include that event, not rolling A four and divide it by the number of total possible outcomes in this case, still six. Now looking at these, if I were to take the probability of A and the probability of its complement and add them together. I see that I get 6/6, which is just one. Now, this makes sense, right? Because we've covered 100% of the possibilities of rolling a six sided die rolling a 12356 or a four. So it makes sense that the total probability of all possible events is simply one. Now, this is always going to be true. The probability of some event plus the probability of its complement is going to be equal to one. And we can use this formula over here to more easily calculate the probability of something not happening by rearranging a little bit here. So if I were to subtract the probability of a from both sides here, it will cancel on that left side, leaving me to see that the probability of the complement of A is equal to one minus the probability of A. So here I see that the probability that something does not happen is simply one minus the probability that it will happen. So now that we know this formula, let's apply it to another example here in this example, I'm asked when drawing a single card from a standard deck of 52 what is the probability that I will not draw a queen? Well, instead of trying to find all of the cards that are not a queen, let's just consider all of the cards that are queens. So if I look at the probability of getting a queen, I know that in a standard deck of 52 cards, there are four queens. So I take all of the outcomes that include my event, drawing a queen and put that over the number of total possible outcomes. In this case, since I have 52 total cards, my total is 52. Then to find the probability of not drawing a queen, I can simply take one minus the probability of drawing that queen, which we just calculated. So we can go ahead and plug in that 4/52 here. Now, I know that one is simply the same thing as 52/52 just getting a common denominator here. So if I perform this subtraction, I end up with 48/52 which as a decimal is 0.92. So the probability of not drawing a queen is 0.92. And we found that by simply using the probability of drawing a queen without having to count up those 48 cards. So now that we know how to find the probability of something not happening, let's get some more practice. Thanks for watching. And I'll see you in the next one.

6

Problem

Problem

When drawing a marble out of a bag of red, green, and yellow marbles 8 times, a red or yellow marble is drawn 6 times. What is the probability of drawing a green marble?

A

0.025

B

0.125

C

0.25

D

0.75

7

Problem

Problem

A weatherman states that the probability that it will rain tomorrow is 10%, or 0.1, & the probability that it will snow is 25%, or 0.25. What is the probability that it will not rain or snow?

A

0.35

B

0.65

C

0.75

D

0.90

8

concept

Probability of Mutually Exclusive Events

Video duration:

5m

Play a video:

Hey, everyone so far we've been dealing with the probability of individual events, like say the probability of picking a blue shirt out of my closet to wear on a given day. But what about the probability of multiple events and further than that, what about multiple events that can happen at the same time? Like say, wearing a blue shirt with a green pair of pants versus those that cannot like say wearing a blue shirt and wearing a green one too. Well, this might sound like it's going to be complicated, but here I'm going to break it down for you and show you exactly what the difference is between events that can happen at the same time versus those that can't so that we can calculate the probability of those that can't. So let's go ahead and get started here. Now, events that cannot happen at the same time actually have their own special name and they're referred to as being mutually exclusive. And you might see multiple events represented using a circle diagram like this one that we see here. Here we have event A wearing a blue shirt and event B wearing a green shirt. Now, in this circle diagram, you might notice that these are two completely separate circles. I have event A and event B but there's no region of overlap because these events are completely separate. They cannot happen at the same time. It's either wear a blue shirt or wear a green shirt but never both. Now because of that these events can't happen at the same time. They're referred to as being mutually exclusive, they're completely exclusive from one another. Now over here, we have event A as still wearing a blue shirt, but we have event B as wearing green pants. Now, in this circle diagram, we see this region of overlap in the middle and this represents both of these events happening together wearing a blue shirt with a green pair of pants. So because of this overlap, we see that these events can happen at the same time and that means that they are not mutually exclusive. Now, we're going to have to differentiate between sets of events that are mutually exclusive versus those that are not. So let's get a bit more practice with that now that we know what mutual exclusive exclusivity even is. So looking at this set of examples here, I see getting heads when get, when flipping a coin versus getting tails. Now whenever you flip a coin, we know that there are only two things that could happen. I could either get heads or I could get tails, but I can never get both at the same time on a single coin flip. Now, because these can't happen at the same time. These are mutually exclusive, they're completely exclusive from one another and their circle diagram would look similar to this one up here. Now, let's look at our second set of events here. We have getting a six when rolling a die versus getting a number higher than three. Now, let's think about this a little bit. When we roll a six, there's only one way to do that by rolling A six. But if we want to roll a number higher than three, we could get a two. I'm sorry, we could get a four, a five or a six. There are three possible outcomes here. Now, one of those outcomes is getting a six, which is exactly what my other event is. So if I were to roll a six, I would be satisfying both of those events and they would be happening at the same time. Now, because of that, these events are not mutually exclusive. Now, here we want to dive deeper into our events that are mutually exclusive and calculate the probability of some event A or some event B occurring. So let's go ahead and dive deeper into that here. So to find the probability of one of multiple mutually exclusive events occurring, all we're going to do is add the probabilities of each of them. So I'm going to take the probability of event A and add it together with the probability of some event B. And that's all this will give me the probability of my two events A or B occurring. Now, this is sometimes referred to as, or probability because it's A or B but never both, right? Because they can't happen at the same time. And you might see this with this little you simple, that just means or in set notation. So that we've seen this formula, let's go ahead and apply it to an example. Here we roll a six sided die and we want to know the probability of getting a three or getting a five. So I have two events here, I have rolling a three or rolling A five. So here I want to calculate the probability of rolling A three or rolling A five. So with this color coded here, we see our two events three or five. And in order to calculate this probability, we saw that we just need to go ahead and add these together. So I'm gonna take the probability of rolling a three and then I'm simply going to add it together with the probability of rolling A five. So we just need to look at these individual probabilities to come to our final answer here. So I know that whenever I roll a six die, there's only one way that I could roll a three by just rolling that right. So there's one way to get to that outcome. And there are six total possible outcomes here because it's a six sided die. Now, I'm going to take that probability and add it together with my probability of rolling a five. Now, I know that there's also only one way to roll a five here. So this probability will actually end up being the same. So I have 16 plus 16. Now actually adding these two together gives me a value of 2/6 or as a simplified fraction, one third. Now we know that one third as a decimal is equal to around 0.33 and that would go on and on. So our probability here of rolling a three or rolling a five is equal to one third or 0.33. Now that we know what or probability is and how to calculate it for mutually exclusive events. Let's get some more practice. Thanks for watching and I'll see you in the next one.

9

Problem

Problem

If a single card is randomly selected from a deck of cards, what is the probability of selecting an ace or a king?

A

0.0059

B

0.077

C

0.15

D

0.85

10

Problem

Problem

For two mutually exclusive events A and B, compute $P(A\xe2\u02c6\xaaB)$ if $P\left(A\right)=0.15$ and $P\left(B\right)=0.32$

A

0.048

B

0.17

C

0.47

D

0.53

11

concept

Probability of Non-Mutually Exclusive Events

Video duration:

5m

Play a video:

Hey, everyone, we just learned how to calculate the or probability of two mutually exclusive events by simply adding the individual probabilities together. But what about the or probability for two non mu mutually exclusive events? Well, you may be worried that we're going to have to learn a brand new formula here, but you don't have to worry about that because here, I'm going to walk you through how calculating the or probability for non mutually exclusive events is actually almost identical to calculating from mutually exclusive events just with one extra step added in. So let's go ahead and get started so that then you can calculate the or probability for any events. So looking at this, remember for our non mutually exclusive events, there exists this region of overlap in the middle where both of these events are happening at the same time event A and event B are happening here. Now, in set notation, you'll see this written as A and B with this little upside down you symbol that just means and now, and actually calculating the probability of A or B happening for these non mutually exclusive events, we're going to start out the same exact way we did for our mutually exclusive events. We're going to take the probability of our event A in this case wearing a blue shirt, whether it be with green pants or not. And we're going to add in our probability of event B in this case, wearing green pants. But in adding in our probability of wearing green pants, we already accounted for the time that it got worn with our blue shirt. So, wouldn't we be counting it twice here? Well, we would be counting it twice. So we really just want to add in the time where we're wearing green pants without a blue shirt. But how do we do this? Well, in order to get rid of that extra outcome and not count it twice, we need to subtract the probability of that overlap region, the probability of A and the probability of B in order to only have counted that area once. So in order to calculate the probability of A or B occurring, we're still adding probability of A plus the probability of B, but now we're just subtracting the probability of A and B. Now, this might seem a little bit abstract and overwhelming seeing this equation for the first time. But here we're going to walk through an example together. So let's go ahead and take a look at this example down here and we'll see exactly how this equation works. So here we see when rolling a six sided die. What is the probability of rolling a number greater than three or an even number? So here we have two events rolling a number greater than three or rolling an even number. So when rolling a six sided die, let's think through all of our possible outcomes here. Well, we could roll one but that actually isn't a part of either of our events. We could also roll a two. Now, a two is even, but it's not greater than three. We could also roll a three, which is not a part of any of my events either. Then I could roll a four. Now a four is both even and greater than three. So it's gonna go in this middle region here. Then I could roll a five, a five is not an even number, but it is greater than three. Then finally, I could roll a six which again is both greater than three and even. So it's gonna go in this overlap region here. Now, from here, let's go ahead and get into calculating this probability, the probability that will roll a number greater than three or an even number. Now, in doing this, remember, we're going to take those individual probabilities and add them together to start out here. So we're going to take the probability of rolling a number greater than three. Looking at my circle di I'm here. I see that there are three possible outcomes either a four, a five or a six out of six total possible outcomes. So that's my first probability. Then for my second event rolling an even number I could roll a two, a four or a six out of six total possible outcomes again. So again, we get 3/6 here. But now I need to subtract that region of overlap where my numbers are both greater than three. And even and looking at my diagram here, there are two outcomes in which that would happen rolling a four or rolling A six. So here I'm going to take that two over my six total possible outcomes. Now, from here, all that's left to do is addition and subtraction. So first taking that addition 36 plus 36 is going to give me 6/6 and then I'm subtracting that 2/6. Now subtracting that gives me 4/6 or as a simplified fraction, two thirds. Now as a decimal, if you want to express your probability here as a decimal, this gives us 0.67. So, so the probability of rolling a number greater than three or an even number is equal to 0.67. Now this makes sense, right? Because looking at this number 4/6 and comparing that to our circle diagram, there are four possibilities that are part of both of these events. So a five, a four, a six or A two, those are four things that are either greater than three or an even number, maybe both mi mixed in there and it's six total outcomes. So 4/6 makes perfect sense here. Now, having calculated the or probability for mutually exclusive and non mutually exclusive events, it may seem like there are two separate formulas to remember here, but not really because this orange formula here is actually the most general way to calculate the probability of any two events A or B whether they're mutually exclusive or not. But if they are mutually exclusive, this last term is just going to be zero. So this formula will end up looking like this for our mutually exclusive events. So the equation for the probability of A or B is really the same regardless of mutual exclusivity. But for our mutually exclusive events, the probability of A and B is simply always going to be zero. So now that we know how to find the or probability for any two events, mutually exclusive or not, let's get some more practice. Thanks for watching and let me know if you have questions.

12

example

Probability of Non-Mutually Exclusive Events Example

Video duration:

2m

Play a video:

Hey, everyone in this problem, we're given a data table and we're told that this table shows the outfits of 300 observed people on a given day. And we want to know of one person randomly selected from this group of 300 people. What is the probability that they will either be wearing shorts or wearing a green shirt? Now, this might look overwhelming just because there's a lot of data there. But let's go ahead and interpret it together. So we see that of the people that are wearing shorts, we come to a total of 100 and 88 out of those 300 people that are wearing shorts. Now, our second event is wearing a green shirt. So we have 100 and six total people that are wearing a green shirt and we can see that breakdown. Now, this is actually a great way to visualize non mu mutually exclusive events because we see that this region of overlap here where both of these things are happening, people are wearing a green shirt with shorts. So both of these events happen together. Now whenever we're dealing with events that are non mutually exclusive we know that we need to account for that region of overlap when calculating the probability. So we know that the probability of our events A or B happening is equal to adding those two events together. So taking the probability of A, in this case, we'll refer to wearing shorts as event A and then adding that with the probability of event B in this case, wearing a green shirt and then subtracting that region of overlap. So subtracting the probability of A and B happening just like we see in our table here, we see that people are doing both event A and event B. Now we can go ahead and use our data to fill in the gaps here and calculate our final probability. So for the probability of event A, we know that there are 100 and 88 out of 300 total people wearing shorts. So we want to take that 188 put it over a total of 300. Then we want to add that together with our probability of event B. So we see that there are 106 total people wearing a green shirt. So we can take that 106 and again, put it over our total of 300. Then finally, we want to subtract that region of overlap, the people that are doing both of these events wearing shorts with a green shirt. So we're going to take that number of 89 and put it over our total as well. So 89/300. Now from here, we're just left to do this calculation. So I can go ahead and add these two values together. And that gives me a number of 294/300. That's a lot of people that are doing both of those events, but I need to subtract that region of overlap. So subtracting that 89/300 and doing this subtraction, I end up with a value of 205/300. Still a significant amount of people but not quite so many. Now, we can go ahead and reduce this fraction to 41/60 or if you want to express this as a dec this goes down to 0.68 as our final answer. So of one randomly selected person of this group of 300 the probability that they will be wearing shorts or a green shirt, maybe even both is 0.68. Thanks for watching and I'll see you in the next one.

13

Problem

Problem

A card is drawn from a standard deck of 52 cards. What is the probability that the card is a diamond or a king?

A

0.33

B

0.31

C

0.15

D

0.85

14

concept

Probability of Multiple Independent Events

Video duration:

5m

Play a video:

Hey, everyone so far, we've learned how to calculate the probability of one event or another event happening, like say getting heads or tails on a single coin flip. But what if someone asks you to calculate the probability of getting heads on one coin flip and then getting tails on another coin flip? Well, it turns out that this is actually an entirely different type of probability that you might hear referred to as and probability. Now we're going to have to calculate this and probability. But before we dive too deep into that, we're going to need to talk about what it means for events to be independent from each other. Now, I know that that might sound like a lot, but here I'm going to walk you through exactly what it means for two events to be independent or dependent and then how to calculate the and probability of two events that are independent. So let's go ahead and get started. So when considering two different events, if these two events do not depend on each other, they are referred to as being independent. So in other words, the outcome of one event has no effect whatsoever. On the outcome of the other event. Now, this can see a little bit abstract. So let's take a look at some different sets of events and determine whether or not they're independent. So looking at my first example, here, I have getting tails on the first toss of a coin and then getting tails on the second toss of a coin. Well, what happens in the first toss has no effect whatsoever on what happens in the second toss. So that tells me that these events are independent from each other. What happens in one doesn't affect what happens in the other. Let's look at another set of events here. We have drawing and keeping a blue marble from a bag and then drawing a blue marble again. Well, if I consider what's actually happening here, if I reach into that bag and I draw a blue marble on that first straw and I keep it in my hand and then I go back in the bag and I want to know the probability of drawing a blue marble again. Well, I've already removed one of the blue marbles. So my chances have lessened because of what happened in that first event now because what happens in the first event affects what happens in the second, these two events are not independent, they are dependent because what happens in one depends on what happens in the other. So now that we've seen how to do, determine whether events are independent or dependent. Let's take a closer look at our independent events here. So for two independent events, we're going to need to calculate the probability of event A and event B occurring. Now earlier I said that you might hear this referred to as and probability. And in set notation, you'll see this denoted with this little upside down you symbol that just means and now actually calculating this probability is super simple because all we're going to do is multiply our probabilities together. So we're going to take the probability of event A occurring and multiply it by the probability of event B occurring in order to get the probability of A and B occurring. So now that we've seen this formula, let's go ahead and apply it to some examples that we will work through together. So looking at our first example here, we want to find the probability of getting heads on two consecutive coin flips. So the probability of getting heads and then getting heads again on a second flip. So in order to do this, we're just going to take these individual probabilities and multiply them together. So we're going to take the probability of getting heads on that first flip and then multiply it by the probability of getting heads on the second flip. So looking at these probabilities, I know that there are only two possibilities when flipping a coin, either heads or tails and only one of those is getting heads So here, the probability of getting heads on that first coin flip is 1/2. Now, I'm going to multiply this by the probability of getting heads on the second coin flip, which is actually going to be the exact same thing because again, there are still two possible outcomes getting heads or tails and only one of those is heads. So I'm gonna take that one half and multiply it one by one half again. Now, when we multiply across here, I'm going to end up getting 1/4 or as a decimal 0.25 as my probability of getting heads and then getting heads again. Now, we can actually reason this out because there's not a ton of possibilities here, right? So if I consider the first flip of a coin and the second flip of a coin on that first flip, I could get heads then on my second flip, I could get heads again or I could get heads on that first coin flip and then tails on the second. But I could also get tails on the first flip and then get heads on the second or I could get tails on the first flip and tails on the second. Now, these are the only four possible things that could happen and only one out of these fours is getting heads on two consecutive coin flips. Now, this backs up what we just calculated because we found that there is a one in four chance of getting heads on two consecutive coin flips. Now, let's take a look at another example. Here we have rolling an even number on the first roll of a six side dye and then rolling a three on the second roll. So we're trying to find the probability of getting an even number and then getting a three when rolling this dye. So of course, we're going to take these individual probabilities and multiply them together. So we're going to take the probability of getting an even number and then multiply it by the probability of getting a three. Now when rolling a six sided die, I know that there are six total possibilities. And in order to roll an even number, I could either get a two, a four or a six. And those would all be even numbers. So that tells me since there are three possible ways to get an even number that my probability of getting an even number at all is three out of those six total possibilities. Now I'm gonna multiply this by the probability of rolling a three, which there is only one way to roll a three. So I know that this probability is 1/6. Now, again multiplying across here, this is going to give me 3/36 which as a decimal is simply 0.08. So not a very high chance that I'll roll an even number and then roll a three Now, there's one final thing that I want to mention for, if you ever run into a problem that asks you to find the and probability of more than two independent events. And it's super simple. All you're going to do is keep multiplying all of those probabilities together to find the probability of A and B and C occurring. Just keep multiplying those individual probabilities. Now that we know how to find the and probability of multiple independent events. Let's get some more practice. Thanks for watching and I'll see you in the next one.

15

Problem

Problem

The spinner below has 6 equal regions. Find the probability of landing on yellow for the first spin and not landing on yellow on the second spin.

A

0.11

B

0.22

C

0.66

D

0.88

16

Problem

Problem

The spinner below has 6 equal colored regions numbered 1-6. Find the probability of stopping on yellow for the first spin, stopping on an even number on the second spin, and stopping on blue or red on the third spin.