in this lesson we're going to focus on mutually exclusive events now what is a mutually exclusive event so these are events that cannot occur at the same time so what does that mean let's say if we have two events event a and event B events A and B will not occur at the same time if they do not share any outcomes now what we're going to do is illustrate this with an example let's say we're rolling a six-sided die with a sample space of 1 to six and let's say that event a represents the outcomes 1 2 and 3 and let's say that event B has the outcomes five and six whereas event C we're going to say has the outcomes three four and five now here's the question A and B are they mutually exclusive events what would you say what is the sample space of A and B what is the intersection of these two events notice that they do not share any outcomes so there's no intersection therefore the probability of A and B occurring together is zero this cannot happen let me write this as a and b because you might think it's a then B so make sure you understand this the probability of two mutually exclusive events occurring is always going to be zero now what about events A and C are they mutually exclusive notice that A and C they share the outcome three so the intersection of these two events is three so the probability of getting a and C is going to be this one outcome over the six possible outcomes so because the probability of getting a and C is not zero A and C are not mutually exclusive events however A and B they are mutually exclusive events now what about events B and C are they mutually exclusive B and C they share the outcome five so that's the intersection of these two events so that's the probability of getting b n c is going to be the same it's 1/ 6 so because it does not equal zero we could say that B and C are not mutually exclusive events so that's how you can tell if two events are mutually exclusive or not you need to find a probability of getting those two events if it's zero they are mutually exclusive if it's anything but zero they are not mutually exclusive now let's illustrate the concept of mutually exclusive events using vent diagrams so let's say this is event a and on the right event B so which V diagram would you say represents mutually exclusive events is it the one on the left or is it the one on the right now we know that mutually exclusive events do not share any outcomes so the one on the right are mutually exclusive events the one on the left are not mutually exclusive events because they do share something in common now how can we calculate the probability of getting event a or event B for something that's not mutually exclusive this is the same as a in Union with b and it's equal to the probability of getting event a plus the probability of getting event B minus the probability of getting both A and B that is the intersection of A and B so this equation works for events that are not mutually exclusive now that doesn't mean it doesn't work for events that are mutually exclusive because it does keep in mind for a mutually exclusive event or events rather the intersection of A and B will be zero so therefore the probability of getting a or b for two mutually exclusive events is simply PA a plus PB if you use this equation this will be zero and it will still work it simply simplifies to the equation that we have on the right if you have two mutually exclusive events now let's work on some example problems let's use our favorite six-sided die with a sample space of the natural numbers from 1 to six now let's say that event a represent the outcomes 1 2 3 and 4 and let's say that event B represent the outcomes three four and five what is the probability of getting a or b by the way are these events mutually exclusive we could see that they're not because they share the outcomes three and four so let's represent this with a vend diagram so because they're not mutually exclusive we need to draw this way this is going to be a and this is going to be B now A and B have the numbers three and four in common so we're going to put that in the middle a also has 1 and two B has five the probability well let's write the formula for this first so P of A or B is going to be P of a plus P of B minus the probability of getting both now what is the probability of getting event a there's four outcomes that relate to event a out of a total of six potential outcomes so it's going to be 4 over 6 now what about the probability of getting event B notice that there's streight outcomes that relate to it so it's going to be 3 out of six if we were to ignore this part of the equation notice that if we add these numbers this would be seven out of six and this is not possible the probability of any event occurring cannot be more than one it can't be greater than 100% so that's why this part of the equation has to be there we have to deduct the events that occur in common otherwise we'll be counting three and four twice so the probability of A and B occurring represents two outcomes out of a total of six so it's 2/ 6 so if we add four and 3 which is seven and then subtract that by two we get 5 6 so that's the probability of event A or B occurring now let's see if this answer makes sense so what is the union of A and B what is the sample space for that a has the numbers 1 to 4 B has the numbers 3 4 and an additional number five so the sample space of A or B is the numbers 1 through through five so to calculate the probability of a in Union with B using this we have five favorable outcomes out of a total of six so we could see why the probability is going to be 5 over six so hopefully this equation makes sense now you see why it works the way it does now let's add a new event in this problem we're going to add event C and C is going to have the outcome six what is the probability of getting B or C so let's write out the formula this is going to be the probability of getting B plus the probability of getting C minus the probability of getting B and C are these events mutually exclusive what would you say notice that B and C do not share any outcomes so this is a mutually exclusive event B and C so if we were to draw a vent diagram it would look something like this this would be B this would be C B has the numbers 3 4 5 C has a number six so these two circles should not intersect because there's nothing in common the probability of getting B is going to be the three outcomes that it has out of six potential outcomes the probability of getting C is one outcome out of six now the probability of getting both B and C that's going to be zero because there's nothing in common so for the entire event is 4 over 6 which is 2 over 3 so anytime you have a mutually exclusive event the probability of those events occurring will always be zero so keep that in mind so thus we have this formula P of B or C is just P of B plus P of C if it's mutually exclusive