MDM 4 U 1 5 Independent and Dependent Events

okay folks looking at 1.5 independent and dependent events compound events involve more than one event for given trial of probability experiment that may or may not affect each other individual events are situations in which the occurrence or non-occurrence of one event has no influence on the probability of the other event occurring multiplicative principle which is known as the fundamental counting principle is as follows for independent events independent event is the probability of two independent events and occurring is probably and the probability of is equal to the probability of time's the probability of now let's go back so this formula is when there are two independent events okay they happen individually and they don't rely on each other for it to happen now the events are dependent when the occurrence or non-occurrence of one event influences the probability of the second event occurring which is conditional prop which conditional probability of second event occurring given that first event occurred that's known as the conditional probability the sample space for the second event is reduced from the first event so multiplicative Purse principal for dependent events is as follows the probability of or and is equal to probability of time's the probability of provided that you've removed so there is condition of happening so the key here is that given that first event occurred the sample space for the second event is reduced the first event so provided that occurs you can then calculate the probability of again this tells you that so probability of two dependent events will occur if you take the product of the probability of the first event multiplied the conditional probability that occurs after has occurred so note that this definition is very important to calculate so we'll look at an example of this and in little bit example number one what is the probability that family will have four female children so if we do tree diagram because easiest to look at it from tree diagrams so first child is gonna be boy or girl second child boy or girl boy or girl third child boy or girl boy girl boy girl boy girl and finally the fourth child is boy girl boy girl boy girl boy and so this tree diagram will allow us to calculate the probability of having four girl children well this is the only way we could have four girl children right here so that there's only one possibility out of total of that's right one out of now you're gonna see like one possibility out of total of all of this and you'll see why you don't need to draw an entire tree diagram to calculate 1 out of 2 4 6 8 10 12 14 16 1 out of 16 can prove this by using that formula we just learned that is we to get the probability of four girls we say okay we have one and two chance of getting girl the first time times one over two of getting girl the second time multiplied by one out of two of getting girl the third time multiplied by one out of two to getting girl the fourth time so finally the probability of getting four girls is equal to one out of that's right sixteen so instead of doing tree diagram which folks imagine trying to do that with cards wow that would be very hard tree diagram to read it is much easier to use that multiple could multiplicative principle and multiply the chances of each individual event happening which is one sixteenth look at another example example number two three Queen marbles and two yellow 3 green marbles sorry and two yellow marbles are placed into bag what is the probability of randomly choosing or drawing green male bro followed by yellow marble assuming that the first marble is replaced before the second marble is drawn the key is it is replaced before the second marble is drawn so you go into the bag and what's your chances of getting green and then finally yellow provided that the green is replaced now this is important we have to replace it because it will if we didn't replace it that will change our total number of marbles so the probability of getting green is how many green marbles there are three green marbles out of total of five now three out of five to get green we've picked the green we put the green back in the bag and now we choose the marbles again and we have two yellow marbles two out of five so to get yellow the second time is two out of five so our chances of getting green and then yellow immediately after will be 6 out of 25 all right now if we do tree diagram imagine we would have 25 possibilities 6 7 the 25 would be green and then followed by yellow alright next example three what is the probability of flipping heads with fair coin enrolling an even number with fair die now these two do not have anything to do with each other so it's very easy the probability probability of heads and even is count we calculate the probability of the heads which is one and two times the probability of getting an even number which is one-half or that's right we can do it three six and then reduce it later or we can reduce it now before we get the answer 1/2 times 3 6 is going to be 1/4 so you have 1 in 4 chance of getting heads and rolling an even number alright next example example number four bag contains two apples one orange and two peaches suppose jenna reaches in and chooses piece of fruit at random and then selects another piece of fruit without replacing the first one what is the probability she will choose two peaches that's the question we need to ask ourselves what's its probability of choosing peach and then peach without replacing not replacing it this is important to understand so what's the chances of picking peach well if we pick peach means that we will have one two possible peaches out of total of five fruit altogether next if we don't replace it how many peaches would we have left suppose we had chosen the peach the desired outcome is choosing the peach we got the peach how many teachers do we have left that's right one one out of total of yes you got it four so the idea is we want to choose peach and we don't care if we chose the other possibilities we want to know the probability of getting peach so that removes one peach out of the bag and to pick another peach with the remaining fruit in the bag so two out of five times one quarter two fifths times 1/4 is two sorry No one second ago some of you may have thought that the answer to the next part was two quarters it's not folks because our desired outcome is to get peach the first time we have 2/5 chance of getting peach the first time if we do get that peach which is what we want then the remaining one is one peach left in the bag out of four so it would not be these values so you have to be really careful to understand what it's asking we want to peach the first time we take that peach out of the running we have four fruit left over and only one of the peaches left over it because we just had the first one let's use Venn diagram to better explain it so in the first time the first chance we have let's use yellow for peaches red for apples and orange for the orange oranges this is our first bag we have two peaches two apples one orange so the second tough so we have two and right now the confusion comes in this piece right here folks is it one quarter or is it two quarters well folks it is one quarter and let me show you using diagram so in my first bag I'm gonna have the fruit and then have my second bag with the remaining fruit so here's my two peaches in yellow to red apples and one orange and in the second bag second time so the first probability is this this is in my bank want peach so have two peaches in total of five that's where the two fifths come from now the second time pick from this bag I've eaten one of the peaches because wanted peach so pulled out peach from the bank what's left over one of the peaches is gone because pulled it out and I'm left with peach two apples and an orange what's my chance of getting peach now in this new bag after the peaches pulled out that's right one in four so take 2/5 times 1/4 and that gives me grand sum total of 2 over 20 which is reduces to 1/10 1/10 of the remaining have 10% chance of getting peach and then peach again after ate the first peach okay example 5 lungs is offering juice samples at shopping mall the experimental probability of randomly chosen shopper accepting sample is 15 percent the conditional probability of customer purchasing some juice given that he or she has tried sample is 20% no one purchases juice without trying sample if Flores offers 500 people juice samples how many sales will he make so the question is how many sales will he make all right so we need to understand Laurence is offering juice samples how many juice samples does he offer he offers 500 people juice samples now how many those people actually accept the sample well only 15% accept the sample well noting that there is conditional probability that if perp customer purchased the juice if he or she has tried the sample is 20% so what we need to calculate how many people action by the juice that Lars is offering so he can meet let's say quota well the probability first we're going to calculate the probability of somebody buying the juice well that is probability of somebody taking sample and buying the juice okay not just but getting sample but also buying the juice well what we note that we have 15% of the people actually take sample 15% of the people take sample and we multiply that but by how many people buy the juice so 20% by the juice so 15% times 20% will equal total of 0.03 that is sample and buy so 15% we'll take sample and 20% of those who tried the sample will buy the juice so 15% times 20% gives you 0.03 so that's in in percentage-wise that's 3% of the people actually buy the juice so Lawrence offers this to 500 people so we need to know the number of people that will sample and buy well to sample and by while he offered sample to 500 people and we calculated the probability of sample and buy and that will be 0.03 which equals 15 people so of the 500 he offered sample to 15 we'll come back and actually buy the juice that Loras is trying to sell all right that's the end of the video folks have numerical day