Hey everyone. We just learned how to calculate the or probability of 2 mutually exclusive events by simply adding the individual probabilities together. But what about the or probability for 2 non mutually exclusive events? Well, you may be worried that we're going to have to learn a brand new formula here, but you don't have to worry about that because here I'm going to walk you through how calculating the or probability for non mutually exclusive events is actually almost identical to calculating it for mutually exclusive events, just with one extra step added in. So let's go ahead and get started so that then you can calculate the or probability for any events. So looking at this, remember for our non mutually exclusive events, there exists this region of overlap in the middle where both of these events are happening at the same time. Event A and event B are happening here. Now in set notation, you'll see this written as A and B with this little upside down U symbol that just means AND. Now, in actually calculating the probability of a or b happening for these non mutually exclusive events, we're going to start out the same exact way we did for our mutually exclusive events. We're going to take the probability of our event a, in this case wearing a blue shirt, whether it be with green pants or not, and we're going to add in our probability of event b, in this case, wearing green pants. But in adding in our probability of wearing green pants, we already accounted for the time that it got worn with our blue shirt. So wouldn't we be counting it twice here? Well, we would be counting it twice. So we really just want to add in the time where we're wearing green pants without a blue shirt. But how do we do this? Well, in order to get rid of that extra outcome and not count it twice, we need to subtract the probability of that overlap region, the probability of a and the probability of b, in order to only have counted that area once. So in order to calculate the probability of a or B occurring, we're still adding probability of A plus the probability of B, but now we're just subtracting the probability of A and B. Now this might seem a little bit abstract and overwhelming seeing this equation for the first time. But here, we're going to walk through an example together. So let's go ahead and take a look at this example down here. And we'll see exactly how this equation works. So here we see when rolling a 6 sided die, what is the probability of rolling a number greater than 3 or an even number? So here we have 2 events, events, rolling a number greater than 3 or rolling an even number. So when rolling a 6 sided die, let's think through all of our possible outcomes here. Well, we could roll a 1, but that actually isn't a part of either of our events. We could also roll a 2. Now, a 2 is even, but it's not greater than 3. We could also roll a 3 which is not a part of any of my events either. Then I could roll a 4. Now a 4 is both even and greater than 3 so it's going to go in this middle region here. Then I could roll a 5. A 5 is not an even number, but it is greater than 3. Then finally, I could roll a 6 which again, is both greater than 3 and even, so it's going to go in this overlap region here. Now from here, let's go ahead and get into calculating this probability, the probability that we'll roll a number greater than 3 or an even number. Now, in doing this, remember, we're going to take those individual probabilities and add them together to start out here. So we're going to take the probability of rolling a number greater than 3. Looking at my circle diagram here, I see that there are 3 possible outcomes, either a 4, a 5, or a 6, out of 6 total possible outcomes. So that's my first probability.
p ( X > 3 ) = 3 6Then for my second event, rolling an even number, I could roll a 2, a 4, or a 6 out of 6 total possible outcomes again. So again, we get 3 over 6 here. But now I need to subtract that region of overlap where my numbers are both greater than 3 and even. And looking at my diagram here, there are 2 outcomes in which that would happen, rolling a 4 or rolling a 6. So here I'm going to take that 2 over my 6 total possible outcomes.
p ( X = even ) = 3 6 p ( X > 3 & X = even ) = 2 6Now from here, all that's left to do is addition and subtraction. So, first taking that addition, 3/6 plus 3/6 is going to give me a 6/6. And then I'm subtracting that 2/6. Now subtracting that gives me 4/6 or, as a simplified fraction, 2 thirds. Now, as a decimal, if you want to express your probability here as a decimal, this gives us 0.67. So the probability of rolling a number greater than 3 or an even number is equal to 0.67. Now, this makes sense, right, because looking at this number, 4 over 6, and comparing that to our circle diagram, there are 4 possibilities that are part of both of these events. So a 5, a 4, a 6, or a 2, those are 4 things that are either greater than 3 or an even number, maybe both mixed in there. And it's 6 total outcomes. So 4 over 6 makes perfect sense here. Now having calculated the or probability for mutually exclusive and non mutually exclusive events, it may seem like there are 2 separate formulas to remember here, but not really, because this formula here is actually the most general way to calculate the probability of any 2 events, a or b, whether they're mutually exclusive or not. But if they are mutually exclusive, this last term is just going to be 0. So this formula will end up looking like this for our mutually exclusive events. So the equation for the probability of a or b is really the same regardless of mutual exclusivity. But for our mutually exclusive events, the probability of a and b is simply always going to be 0. So now that we know how to find the or probability for any 2 events, mutually exclusive or not, let's get some more practice. Thanks for watching, and let me know if you have questions.