Multiplication Addition Rule Probability Mutually Exclusive Independent Events

now we're going to talk about two basic rules of probability the first one is sometimes refer to as the addition rule the probability of forget an event or event can be written as the probability of and union with this is equal to the probability of event no current plus the probability of event occurring minus the probability of event and event occurring together so make sure you know this now sometimes you may be dealing with two events that are mutually exclusive for mutually exclusive events the probability of get an event and together is zero so therefore the probability of getting or becomes of plus of if and are mutually exclusive events but you can always use this formula which will always work regardless of what type of events you're dealing with the next type of rule is the multiplication rule so let's say if well first let's talk about conditional probability the probability of given is the probability of getting and divided by the probability of getting so what this means is that if we multiply both sides by of we can get this equation the probability of getting and is the product of the probability of getting given times the probability of event occurring now what about this scenario what is the probability of given this is still the probability of and occurring but divided by the probability of so if we rearrange this equation by multiplying both sides by of and we see that the probability of and can also be written as the probability of given time's the probability of so notice the similarities of these two equations now sometimes the two events that you're dealing with might be independent events an independent event is an event that does not depend on another event for independent events the probability of given is equal to the probability of event occurring because does not depend on likewise the probability of given is equal to the probability of event occurring because does not depend on so when dealing with independent events we get this equation the probability of and occurring is simply the probability of times not plus but times the probability of event occurring if the events are independent if not then you can get one of these equations depending on what you were given one thing do want to mention is that when you see and occurring this means that these two events are occurring at the same time it doesn't mean and then which could be different from and then so because the events occur at the same time the order is not relevant otherwise the formulas may be affected so just keep that in mind therefore the probability of and occurring is the same as the probability of and occurring because these two events occur at the same time so just want to add that clarification for those of you who may have questions on it now let's go ahead and put this information to good use Sarah is deciding which courses she wants to take in her next college semester the probability that she enrolls in an algebra course is 0.3 and the probability that she enrolls in biology course is 0.7 the probability that she will enroll in an algebra course given that she enrolls in bio course is 0.4 so Part what is the probability that she will enroll in both an algebra course and biology course so take minute pause the video and using the formulas that we discuss go ahead and get the answers to these questions so first let's begin by writing down what we know the probability that she takes an algebra course that is of is 0.3 zero and the probability that she takes bio course that's 0.7 zero the bility of given that is that she takes an algebra course given that she takes bio course that's point four zero so with this information we can now focus on Part so we're looking for the probability that she will take both algebra and bio so what's the formula for this this is equal to the probability of given times the probability of either occur so we can see that of given is 0.4 0 and the probability that she will take bio course is 0.7 0 times 7 is 28 so 0.4 times point 7 is point 28 so there's 28% chance that she's gonna take both algebra and biology now let's move on to Part what is the probability that she will enroll in an algebra course or biology course so this time we're looking for of or so what's the formula for this so this is going to be based on the addition rule as opposed to the multiplication rule so this is of plus of minus of and now the probability that event will occur is 0.3 and probability that she's gonna take the biology course is 0.7 the probability that she will take both algebra and biology we have it here that is 0.28 so let's go ahead and subtract these numbers so 0.3 plus 0.7 minus 0.28 this is equal to 0.72 so there's 72% chance that she will take algebra or biology now let's move on to Part are the two events independent what would you say what are the requirements for two events to be independent in order for events and to be independent the probability of given must be equal to the probability of are they equal the probability of given we could see that it's point four and probability of event occurring is 0.3 so these two are not equal therefore the two events are not independent of each other now what about Part are the two events in mutually exclusive what would you say in order for the two events to be mutually exclusive the probability of event occurring and event occurring must be equal to zero is this true well we know that and the probability of these two events occurring is point two eight because that's what we calculated in Part so it does not equal zero which means that events and are not mutually exclusive events