Compound Probability and Independent Events

George Ingersoll
A free video tutorial from George Ingersoll
MBA & PhD
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Lecture description

How to find the probability of multiple events all taking place when we know the probability of each event.

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Workshop in Probability and Statistics

This workshop will teach you the fundamentals of statistics in order to give you a leg up at work or in school.

21:37:01 of on-demand video • Updated April 2020

  • By the end of this workshop you should be able to pass any introductory statistics course
  • This workshop will teach you probability, sampling, regression, and decision analysis
English All right in this video we're going to talk about compound probability and independent events. Let's begin by defining compound probability this is a fairly simple concept compound probabilities means there are multiple events and we're trying to find the intersection of those events. Meaning if we get two events A and B we're trying to find the probability of occurring and B occurring . So that's that's the intersection we're saying was probability of both events occurring. Now on the surface making this calculation is fairly simple. All you're really going to do is you're going to multiply these two probabilities together. The challenge is that sometimes you're going to have to make an adjustment to those probabilities because they have some sort of an influence on each other. Not all the time it has. Sometimes you can just multiply the two probabilities as if you know they were happening on their own together and you'll get the intersection of those two events that sometimes some of the times you don't have to make an adjustment. And this all has to do with whether the two events are dependent or independent. And we can define this is saying dependent events have some influence on each other and write down these two terms because these are important dependent events do influence each other's outcomes and independent events do not. Meaning if an event occurs it has no bearing on event. B let's start off by talking about independent events because they're actually the simpler type events to deal with when we're talking about compound probability. Let's think about some examples of independent. Well if I have a six sided die and I roll it once and then I roll again whatever I do I will call the first roll event a whatever I got on my first roll has no bearing on what I got my second roll. Event B. So there is no relationship between those two events. You're not going to make a bet on the second roll of a die based on what you got on the first roll they have no bearing. So those are independent events. Roll one event A and roll two of it be independent. Another example let's say I draw a card from a deck of 52 cards and then I put that card back and I shuffle the deck multiple times and I draw. In that case whatever I've got on my first draw does not influence what I call my second or those are independent events. I mean if I get seven of diamonds on my first draw and I put it back and I shuffle everything the chance of me getting seven and diamonds on my second drawing is the same as me getting any other card. So again independent events seem to be true just don't match that that dice idea. So that were totally clear. If I had two different dice that they had red dye and white die then I have to be different colors. But I'm just just for the sake of simplicity let's say I've got a red white dot. I roll them both together. Whatever I get on each die that those are going to be independent events. So I was like that how you would do this calculation when you're talking about compound probability you're always going to do multiplication you're always going to multiply the chance to get. The key is with intuitive independent events. You don't have to make any sort of adjustments to the probabilities and you multiply that with deep and events. You do the equation. This is very simple for independent events to find their intersection and compound probability probability of A and B where the intersection of A and B same thing is the probability of event A Times probability of event B for independent events don't have to make any adjustments you can just take the straight probabilities and multiply them together. Let's look at this in the context of the dice we're talking about. So a red dye and a blue. When I say white dyed red die. So if we said event A is to roll a three on the white die and Event B is to roll a 6 on the red then the probability will think about this in terms of our fundamental probability equation which is you know the number of outcomes we're looking for over total possible outcomes. In this case what's how many outcomes are we looking for. Well we have two or three on a white and a six on the read that's only one possible outcome. That's the only way this can shake out where we're getting what we're looking for. But how many total possible outcomes do we have now more than just a roll on a single dime on a single dime we have sex but in it now that we're we have a compound probably down to the business. We actually have six times six total possible outcomes. We've got we can rule any one of six numbers on the first in any number one any one of six numbers on the second. So essentially for if we say OK this is white and this is red We've got 1 2 3 4 1 around and around 5 6. And if you roll one on point you've got all these possibilities for rolling something I'm red you've got one two three four five six on red and the same thing would be true if you will to one white three on white for it all the way down. So it's six by six. So we have 36 total possibilities. So our probability of A and B here is going to be equal to probability of rolling a three on white which is one sixth times probability of rolling a six on red which is one sixth as well equals 1 out of 36 . That's the slightly different example I'm going to race a little bit of this . Now I'm going to define a new events. We're going to say see is roll even on red. So now we have and we're going to look for the probability of a intersections see think about this. What's our probability of rolling it even on that all the even numbers on a six sided 2.6. So we're still going to use the probability of eight times the probability to be probably still three rolling three. I'm so excited at this that one sixth time is probably wrong. Even on red This one has three possible outcomes we're looking for out of six. So now our probability of a intersection C is three out of 36 and that's what you do for independent . That's these are independent events that don't influence each other. So it's it we can just multiply straight up. That's all we have to do. One more example having to do with dice. I want to take a look at the probability of rolling between the two dice getting six. So you know the two numbers together are adding to six. So probability of rolling a six on two days that this is a little bit of a different situation we still have the same total possible outcomes which is six times six. This is something you're always in use when you when you've got two independent events and you're multiplying them together when you have the intersection you multiply that the sample spaces by each other to get the total possible outcomes. We still have 36. We know the probability of rolling a six on today is going to be 36 as well. But the numerator What is the number of outcomes we're looking for. For this we have to sort of think a little bit hard. We have to think how many ways can you get a six using a red and white. Well let's let's put this down. We've got a white guy here got a red dye here. What are the combinations we kind of well we can have a one in a five. We can have a two and four we can have a three and a three or we can have 4 and 2 or 5 and a 1. Those are one two three four five possible outcomes for rolling a six on two dice and our denominator is a total possible outcomes that we get by multiplying six possible outcomes. Time six possible outcomes. So that's our answer for the probability of rolling on two dice starting to pace. Now you move on to another example let's say in a certain state we've got a certain way of putting together license plates This is the most uneven digital spaces I possibly could have made but just bear with me. There are seven spots for numbers and letters on this state's license plate seven spaces and you have to have a certain number of digits 0 through 9 and letters so 26 letters in the alphabet and they have to be in certain spaces so space one has to have a digit Space 2 3 and 4 have to have letters and then we're going to have another digit digit and another digit. I don't know if this makes any sense or not but you understand what I'm saying here. This is the regulation for the license plates in this state. Completely made it up. You have to have a digit in the first place followed by a letter a letter a letter and then three more digits. So my question for you is how many different possible combinations of license plates are there for this day. Well you could get that you say these events are all independent. You know you could assign a random let's say these are the side randomly so you sign up a random digit to the first place where you assign a random letter here. Another round letter here and the written letter here. Did you did it did. The answer is you have 10 possibilities here. 0 3 9 that's 10 possibilities times 26 possibilities times 26 possibilities times 26 times ten times 10 times 10 equals that equals a big number. I don't know what it is off the top of my head you can do it on your own. You multiply it's the product of all these possibilities. That's the number of possible combinations you can have. And if you want to figure out the probability of getting any specific combination of digits and letters it's going to be one over this number one over what the product of all these is. I'm not going to open up excel and do this. You could work it out on your own. All right here's a twist. What if you can only use one a letter and a digit one time meaning there can be no repeats. It can no repeats in this sequence of digits and letters in your license plate. That significantly complicates things. Let's think about this. OK. In the first space I recreate this hopefully better this time. I think those marginally better or in the first place you can have 10 digits right. Not impossible to have a repeat. But thus far we haven't we haven't decided yet. And same for the second place we have 26 letters because you know there is no possibility of having had a repeat at this point. But in the third place we can only have 25 possibilities because we've already signed a letter in the second place and in the fourth place we can only have 24 possibilities because we've had a letter in the second place alone the third place and now we only have 24 letters to choose from. Similarly in the fifth spot we can only have nine digits and Six-Party can only have eight. And then by the seventh spot we've already assigned one to three digits. We can only seven and figure out the total possible combinations here. We want to multiply these together so this example of events that are not independent they actually do play a role and influence each other because we've we weren't replacing essentially we were saying you know it's we can't have repeats. So whatever you picked in spot two is going to influence your possible outcomes in this spot three. So this is a fairly simple one to wrap your head around and say Well look I know how many possible outcomes I've gotten into these spots. So you know if I'm looking for the total possible outcomes the intersection of all of these events I want to say what's the probability of me getting one surprise at what is probably me getting you know three J G I 2 6 9. Just what are the random random chance of me getting that specific license plate. Well it's going to if you know you can't have repeats it's going to be one over this. All right let's do one more example. I've got time for it. What if I've got a socket or I should let me have it up and this new page here. The sock drawer and in that sock drawer. I've got eight pairs black socks five blue and seven whites and say open up that south towards dark in my room I reach a hand and I'm also not looking I run around pull up our socks. What's the probability of that those socks being black. Well this should be fairly easy for you now this is just a probability of me pulling a black sock. Well probability is going to be equal to number of possible outcomes we're looking for. Eight were total possible outcomes. If you got these up gives you 20 20 also known as Two fifths or 40 percent 40 percent chance pulling out a white sock. Right. What if I did the exact same thing. Let's all Roland transactor pull out of south. Look at it. Say I turn on the lights for a moment and say OK that's a certain house. I turn the lights off and put it back rummaged around and makes everything up. I plot another sort of turn on the lights and I look at that song. What's probably both of the socks that I pulled out would be black. Here we have two events and all the independent. Let's think about this. Does what I pulled out the first time influence but I pulled up a second time. Think about that. The answer is no because I replaced it. I placed the song and then I ran round again I mixed it up again so I pulled out the second time has no bearing on it. I pulled out the first and has no bearing on what it is. So I'll probably then of that so I can multiply them together . This is going to be my first event which we're calling the first poll is going to be it of 20. And second poll is also going to be 20. Probably that and both being black and so that's going to be equal to something like 16 percent. You can work it out on your own. All right here's here's the last twist. This time I'm going to rummage around the sock drawer pull out a sock look at it. But then I can replace it. I'm going to rummage around the soccer again and plot another song. Now I want to know the probability of both of the socks that I pulled out of the door being black. Think about this for a moment. Is this going to be an independent event or is this Are these independent events we're not the answer . Actually you know what I'd really like for you to do is to solve this one on your own. So pause the video in and do this. Figure out what you think the probability of playing both socks out if I don't replace the first one . It was probably both of the black for my pull. I don't pull that. I don't replace the first one. Think about it for a moment. The answer is probably of pulling a black check on the first attempt is of course 020 that hasn't changed the probability of pulling a black sock out in the second attempt. There's going to be seven because that's how many blacks are left at 19 because now I only have 19 possible pairs in the drawer that I can choose from. So that's going to be equal to 56 over 380 which are not going to simplify to them or 20 times even 20 times 19. And so that is an example of dependent events because they influence each other. When I pull down the first time influences I'm going to get the second time. This will be true even if I had said my second pulse can be a red sock I've changed this I've changed from 20 possible outcomes to 219. That's going to affect the probability. One last thing I want to point out and that is this that the equation I gave you earlier for independent events this is this equation is true. A and B are independent can actually be used to check for independence and we're going to look at the next video the next video is going to be up conditional probability conditional probability is dealing with events that are dependent and figuring out the probability of we'll call the second event or another event occurring. If the first one is true. So if we say if a has already happened what's the probability of getting beat for this deep these dependent events. The fact that we got a is going to influence the probability B and it's all about figuring out that probably we've done some independent events like the soccer door that are pretty easy to figure out the probability of B you can get a little bit more complicated sometimes so conditional probability is all about saying OK do we have deep end events here. If so we can use the way of thinking about conditional probability to figure out the probability of B and this compound probability situation we have here and this equation here can actually be a test of independence. OK. Lots to keep in mind. We'll get into this more in the next video.