 This is an example on how to calculate probabilities of independent and dependent events In my example here, I have a bag which contains nine red marbles 10 white marbles and eight blue marbles You draw four marbles out of that random without replacement. So this word without replacement is very important Why because as I draw out each of my four marbles one at a time I am not going to put that marble back in the bag So this means that my events are Dependent so what is the probability that all marbles are red? Well the probability all marbles are red so first drawing I get a red marble Second drawing I get a red marble third drawing I get a red marble fourth drawing I get a red marble I'm going to calculate each of these probabilities and then multiply them together Remember this is being done without replacement. So the total number of marbles I have Will decrease by one each time So drawing number one. I have how many marbles total? 27 marbles 9 plus 10 plus 8 So out of my 27 marbles total how many are red? nine Then we go to draw the second marble out now only have 26 marbles left. How many are red? Well, there were nine, but the first trial I pulled one out. So now there's eight Third drawing there's seven red marbles out of 25 total and Last drawing there's six red marbles out of 24 total Now you can choose to multiply all of these together Multiply the tops together multiply bottoms together or you can go ahead and simplify the fractions 9 over 27 becomes 1 3rd. Why because you divide the top and bottom both by 9 8 over 26 becomes 4 over 13 7 over 25 cannot be simplified and 6 over 24 you divide the top and bottom both by 6 to get 1 4th now 1 times 4 times 27 times 1 is 28 and 3 times 13 times 25 times 4 is 3900 What is 28 divided by 3900 it is point 007 I rounded the three decimal places Part B. What is the probability that exactly two of the marbles are red? So we're looking at all the outcomes of four marbles where two of them are red. So one possibility would be red red not red not red So I'm going to find the probability of each of these Outcomes here. So first trial here Probability of getting a red marble is nine marbles out of 27 Then second marble being red. There's now only eight marbles out of 26 marbles total Third trial or third marble, there's now only 25 marbles to pick from how many are not red Well, there's eight blue and 10 white. So that would be 18 Force trial or force drawing and I only have 17 marbles that are not red out of 24 marbles total I remember I'm having one less marble every single time and I had one less red marble in my second trial Then I had one less not red marble in my fourth trial So you can multiply together your fractions or you can simplify nine over 27 is one third Eight over 26 is four thirteenths 18 over 25 does not simplify Neither does 17 over 24 multiply the numerators together four times 18 times 17 is 1224 Multiply the denominators together three times 13 times 25 times 24 is 23,400 And this is not my final answer, but just the intermediate step to get me there So do not round whatever this is It is actually going to be point zero five two three zero seven But wait, this is just one possible way that you could get two red marbles when you draw out four So the first two could be red last two could not be read Let's list some other outcomes where exactly two of the marbles are red There is red not red not red red There is red not red red not red There is not red not red red red Then there is not red red not red red And the last one is not red red red red not red These are all the different ways you can get exactly two red marbles. Notice there's one two three four five six ways total So what you do is you take this probability we calculated and multiply it by six So six times point zero five two three zero seven Which this will be our final answer. So go ahead and round the three decimal places you get point three one four That is the probability you get exactly two red marbles Last part. What is the probability none of the marbles are red? So we're after not red on the first drawing Not red on the second drawing not red on the third drawing Not red on the fourth drawing Remember without replacement So the total number marbles we have each time will get less the total of not red marbles we have each time will also be fewer So what is the probability we get a marble that is not red on the first trial? Well, that would be 18 marbles that are not red out of 27 total Second trial not red. Well, that would be 17 marbles now out of 26 third trial not red 16 marbles now out of 25 And fourth trial not red 15 marbles out of 24 You can go ahead and multiply your tops together bottoms together Or you could simplify your answer 18 divided by 27 That would be two thirds 17 over 26 does not simplify 16 out of the 25 does not simplify but 15 over 24 That's five over eight Multiply the numerators together Get two thousand seven hundred twenty Multiply your denominators together You'll get fifteen thousand six hundred Divide the top by the bottom And you'll get point one seven four So what we did was we just calculated probabilities of two events or four events. I should say that were dependent Thanks for watching