in this video we're going to talk about two types of events independent events and dependent events now you might be wondering what is the difference between the two as the name suggests independent events are events that do not depend on each other dependent events are events that do depend on each other and we're going to use this example to illustrate this concept so let's go ahead and work out this problem number one a bag consists of eight red marbles seven blue marbles six green Marbles and four yellow marbles what is the probability of selecting a red marble now if you want to work out this problem pause the video and go ahead and do so so let's begin let's write out the marbles that we have so the first color is red the second color is blue the third color is green and the fourth color is yellow so you can clearly see the information now we have eight red marbles seven blue marbles six green Marbles and four yellow marbles now let's get the total 6 + 4 is 10 8 + 7 is is 15 15 + 10 is 25 so we have a total of 25 marbles so what's the probability of selecting a red marble to calculate the probability of an event occurring it's basically the number of favorable outcomes divided by the total possible outcomes in this case it has to do with marbles so it's going to be the eight red marbles that we have ID the total number of marbles which is 25 8 ID 25 is 32 so what this means is that there's a 32% chance of selecting a red marble on the first try now what about Part B what is the probability of selecting a Blue Marble on the first try and then a green marble on the second try with Replacements so what is the probability of getting B and then getting G so let's do this one at a time the probability of getting a Blue Marble is going to be the number of blue marbles which is 7 / the total number of marbles which is 25 now what about the green marble well we need to understand the expression with replacement so that means that once we select the Blue Marble on the first try we're going to put that Blue Marble back in the back so we still have a total of 25 Marbles and there are six green marbles so the probability of getting a green marble will be six out of 25 so now let's do the math 7 * 6 is 42 25 * 25 is 625 now 42 ID 625 that's 0672 so if we multiply that by 100 that corresponds to a 6.7 2% chance of getting a Blue Marble on the first try and then a green marble on the second try now let's move on to part C what is the probability of selecting a yellow marble on the first try and then a red marble on the second try without replacement so what do you think the answer is going to be for that one so let's start with the first event what is the probability of selecting a yellow marble there's four yellow marbles out of a total of 25 so it's four out of 25 now what do you think the expression without replacement means this means that once we take out that yellow marble we are not going to put it back into the back so we no longer have have 25 marbles in a bag but we now have 24 marbles now we still have eight red marbles so to select a red marble on the second try it's going to be 8 over 24 as opposed to 8 over 25 so let's do the math 24 I'm going to write that as 8 * 3 and 8 I'm going to leave it as 8 * 1 so we could can the eight 25 * 3 is 75 so the probability will be 4 over 75 4 ID 75 is 0.53 with the three repeating so this is approximately a 5.3% chance of occurring that's for this particular event now let's talk about what we have here which situation represents an independent event and which one would you say are dependent events so Part B there's two events getting a blue marble and then a green marble would you say the second event depends on the first event what is the probability of getting a green marble just of getting a green marble it's going to be the six green marbles out of the total of 25 and notice that is what we see in the second event the probability hasn't changed so getting the Blue Marble on the first try had no effect on a probability of getting a green marble on the second try so Part B represented a situation with independent events now what about part C the probability of getting a red marble is 8 over 25 did that probability change does the second event depend on the first event yes it does because with without replacement the number of marbles in the back changes and so after you take out that first marble you no longer have 25 marbl you now have 24 and so since the number of marbles in the back changes the probability of the second event will change as you can see the second event is 8 out of 24 compared to what it would be which would be 8 out of 25 if we didn't select the yellow marble on the first try without replacement so anytime you have a situation where it's without replacement you're dealing with dependent events when it's with replacement you're dealing with independent events so keep that in mind now let's move on to Part D what is the probability of selecting two blue marbles with Replacements so is this going to be dependent events or independent events since it's with replacement we're dealing with independent events so the probability of getting the first Blue Marble is 7 out of 25 now we're going to put that Blue Marble back in so we're still going to have a total of 25 Marbles and there still going to be seven blue marbles so on the second try the probability will not change 7 * 7 is 49 25 * 25 is 625 so this is equal to 0784 so there's a 7.84% chance of selecting two blue marbles with replacement now part e what is the probability of selecting two green marbles without replacement so now we're dealing with dependent events so we have six green marbles out of a total of 25 now once we take out one green marble we now only have 24 marbles in the bag now we don't have six green marbles anymore because we took out one we now have five green Marbles and so this is the situation that we now have let's write 25 as 5 * 5 and let's write 24 as 6 * 4 so we could cancel a five and we can cancel a six so since there are no more numbers on top we're just going to put a one on the bottom we have 5 * 4 which is 20 1 / 20 is 05 so if we multiply that by 100 this tells us that there's a 5% chance of selecting two green marbles without replacement