Compound Probability with Independent Events FULL LESSON WITH PRACTICE PROBLEMS AND SOLUTIONS

welcome back to mr. ace math this lesson is on compound probability where we're talking about the events of the probability being independent events some stuff you should know already are the basics of probability but sample spaces are how to use tree diagrams as well as multiplying fractions this mathematically is probably the most important part of the lesson so make sure you know how to multiply fractions and all those other stuff and let's get started what exactly is compound probability well compound probability is basically just regular probability but what you have instead of single event are compound events and compound events are when you have more than one events one after the other and it's the product of each individual event independent events are events that basically the first event has no effect on the second for example let's say we flip coin twice so remember what we said compound probability is we said compound probability is basically just like regular standard probability but instead of having just one event we have more than one event we can have two events we can have three events in this particular example we have two events the first event is landing on heads and the second event is landing on tails but let's deal with heads first what is the probability of landing on heads while the probably landing on heads is 1/2 now our second event is landing on tails ok what is the probability of landing on tails well that's the same exact thing as heads that's also 1/2 and then we have to multiply each of our probabilities and that will give us 1/4 now want to show you exactly why that's one force lot of you just say hey you're multiplied there's the answer good job do it again but want to show you exactly why that is with particular tree diagram here we have tree diagram for what happens if we flip the coin twice this column here represents our first flip and this represents our second flip when we flip the coin the first time it can land on heads or can land on tails now if we land on heads the first time we can still get heads or we could still get tails doesn't really matter same thing for if we land on tails first we can still get heads or we can still get tails for second flip long story short want to get to the sample space this sample space has four outcomes and that's why our denominator is four how many of those outcomes have heads for the first event and tails for the second event well that's this one heads for the first event tails for the second event heads for the first flip tails for the second flip and that's exactly why our probability is 1/4 because of the four outcomes here only one of them has the exact outcomes away we want them heads for the first flip and tails for the second flip now what if we had the same situation we flip coin twice and we want to land on tails for the first event and then tails for the second event well what's the probability of getting tails for our first event well that's gonna be one half and then tails for our second event that is also still one half so then we multiply our probabilities and we end up getting 1/4 now again want to want to take second to look at the tree diagram so you know exactly why that is well our sample space has four outcomes we have one two three four and that's why our denominator is four of those four outcomes which ones does it happen then we get tails for the first event and tails for the second event well that's this one here that's the only one and therefore since that's the only one out of four our compound probability is one-fourth again just want to take second to stress that each of these flips are independent events well what is that word independent mean independent means that it's not affected by if you say that you are independent it means that you're not affected by the actions of anybody else well independent events really mean the same thing their events where the second event isn't affected by the first event so here when we flip our coin the second time it doesn't matter if we land on heads or tails that has no effect on the second flip at all and therefore they are independent events so let's say that we're not talking about corn anymore let's say that we're talking about standard number cube you might hear it referred to as dye or number cube either way it's fine it means the same exact thing so we're saying here that we have two events well what are they the first is the probability of rolling 1 well what's the probability of rolling 1 on standard number cube well there's six sides one of those sides has number one on it so that's going to be 1/6 the second event that we're looking to get is landing on an even number well what's the probability of landing on an even number how many even numbers are there well there's three even numbers two four and six that makes three so that's three out of six now we're gonna multiply those but before we multiply those we can actually reduce to 3 over 6 we're going to divide each of those by 3 and that's gonna reduce to 1/2 and then we just bring down 1/6 times 1/2 so now what do we do well we just multiply 1/6 times one half and that will give us 1/12 and again just want to go over exactly why the probability is 1 over 12 well it's important to know that when we have 6 outcomes that are possible for the first event and there are 6 outcomes possible for the second event to fund this total number of events that are possible we multiply 6 times 6 and that gives us 36 so that means that overall there are 36 possible outcomes that we can have when we when we roll number cube twice so of the 36 how many different ways are there to end up with 1 on the first roll and an even number on the second well there are three we can roll 1 and then we can roll 2 we can roll 1 and then roll 4 and we can roll 1 and then roll 6 now these are our three outcomes but that's going to be 3 over 36 if we reduce 3 over 36 we end up with 1/12 therefore our probability is 1/12 here we have two different events the probability of landing on an odd number for the first roll and the number less than 6 for the second roll well let's talk about the first event probability landing on an odd number well how many odd numbers are there there's three of them three out of six there's one that you can roll three you can roll and five you can roll therefore there are three odd numbers out of total of six now the second event number less than six how many of those are there there's five out of six one is less than 6 2 is less than six 3 is less than 6 4 is less than 6 and 5 is less than 6 so there are five out of six we can roll now we just take our probabilities and multiply them now before we even multiply them we can actually reduce 3 over 6 to 1/2 and bring down the rest so now I've got 1/2 times 5/6 and that's gonna give us 5 over 12 again want to show you exactly why that is remember the total number of outcomes is actually 36 because there are 6 Possible's for the first event and 6 Possible's for the second event 6 times 6 gives us 36 well of those 36 different ways how many ways can roll an odd and then 6 well can roll 1 then 1 1 then 2 1 then 3 1 then 4 1 then 5 can also roll 3 then 1 3 then 2 3 then 3 3 then 4 3 then 5 and can also roll 5 then 1 5 then to 5 then 3 5 then 4 5 and 5 it seems like it's really complicated but you're just saying here that was an odd number for the first outcome and then rolling number that is less than 6 for the second outcome now there's 15 out of 36 since 15 out of 36 can reduce to 5 over 12 our probability is just 5 over 12 how about here we have two events the first event is landing on number that is greater than 4 and what's the probability of landing on number greater than 4 well that's 2 out of 6 because there are 2 numbers on the number cube that are greater than 4 5 is greater than 4 and 6 is greater than 4 and that's where we get 2 out of 6 then not 5 what's the probability of getting number that's not 5 well that's five out of six because you can land on one you can land on two you can land on three you can land on four and you can land on six but you can't land on five so that's only five out of six now that we have our two probabilities we just multiply but we can reduce to over six and that's gonna be one over three and then bring down the rest so now we have one-third times 5/6 and that's gonna be five over 18 again just want to show you exactly what that looks like remember our first outcome needs to be greater than 4 and the second number needs to be any number that's not 5 so what outcomes can be 5 then 1 5 then 2 5 then 3 we can roll 5 and then 4 and we can roll 5 and then 6 we can also roll 6 then 1 6/10 2 6/10 of 3 6/10 of 4 and 6 then 6 now that is 10 outcomes and remember 6 times 6 is 36 so that's really 10 outcomes out of 36 and if we reduce 10 out of 36 we get 5 over 18 and therefore our probability is 5 over 18 how about here we have five tiles and they spell out the word class here we're talking about probability with replacement with replacement is pretty simple it basically means you take out one from the first event and then put it back here's an example let's say we're looking for the probability of drawing the letter and then drawing the letter again so what's the probability of getting an from the word class well there's five total letters and of the five letters as one out therefore it's 1/5 now we basically with replacement means that we take out what we're looking for or one of what we're looking for here we were looking for the letter so we take the letter out but we're replacing it we're placing it again so we just put it right back in the word so now we still have the word class now what's the probability of getting the letter well there's still 1 out of 5 so therefore it's 1/5 so now we have our probability and we'll multiply 1/5 times 1/5 and that gives us 1 out of 25 so our probability is 1 out of 25 here what's the probability of drawing an and then drawing the bowel well let's talk about the first how many S's are there well there's 2 out of how many 5 so the probability of drawing an is 2 out of 5 now we basically take one of the asses out it doesn't matter which one okay and then we're replacing it so we're putting it right back so we still really have the word class so what's the probability of the second event now the probability of getting bowel well there's only one bowel that's the letter out of how many letters five so the probability of getting bowel is 1 out of 5 now we have our probabilities we multiply them and we get two out of 25 and that is our probability how about the probability of selecting an and then letter that's not an well the first event is the probability is 2 out of 5 because there's two S's out of 5 letters so we take one of the esses out again doesn't matter which one and we're saying it's with replacement so we're putting the right back where we got it and that's gonna be class so now we're looking for the probability of second event which is not getting an and getting any letter that's not an well how many letters there are not is just three the three letters are Ln so that's gonna be 3 out of 5 now that have my probabilities multiply them and that's gonna give me 6 out of 25 just because we've been using two events for the last examples doesn't mean you can't have more than two events here we have coin and let's say we flip it three times so we're looking for the probability of landing on heads three times well how does that look like what's probability notation well it's gonna look like this the probability of landing on heads for the first event that's the first event and the probability landing on heads for the second event and then the probability of landing heads for the third event even though it has three events don't want you to feel intimidated we've learned enough that we know all we have to do is find the probability of each of the events separately now remember the three events are landing on heads the first time landing on heads the second time and ending on heads the third time so let's talk about the first one what's the probability of landing on heads the first time well that's one half how about the second time one half and the third time one half and exactly the same way as we've done it in the past we just take our probabilities for each of the individual events and multiply them now let's multiply the first two the first and second events we're gonna have one half times one half and that gives us 1/4 then we bring down the rest that we have in use yet two times one half now we've got 1/4 times 1/2 and all we have to do there is multiply that and now we have 1/8 so that's our probability for flipping coin three times and landing on heads all three times so here's your pause and practice just pause and practice when you're done unpause the video after 3 2 1 countdown your answers will be displayed go okay so let's go over our answers number one the answer is one for number two is 1/12 number three is two ninths number four is 136 and number five is 136 let's review an event that consists of more than one event or parts is called compound event in compound event when the first event has no effect on the second they are called independent events remember that word independent by itself just means not affected by outside factors so independent events are events that don't affect each other to find the probability of compound events you have to blank the probability for each event we simply multiply don't forget to Like share and subscribe questions comments leave them down below and thanks for using mr. ace Matt don't just pass Matt ace it