 and welcome back. Today we're going to talk about inclusive events. So now in a previous video we talked about mutually exclusive events, which exclusive events were events that basically either one of them could happen or the other, and that was basically all the choices you had. Inclusive events actually takes into account that there's going to be a little bit of crossover with the events that could happen. Inclusive events, the probability of two events occurring is the sum of their individual probabilities minus the probability of both of them occurring. Okay, so here's the notation behind that. For two inclusive events, a and b, the probability of a or b happening is equal to the probability of a plus the probability of b minus the probability of a and b happening both occurring at the same time. Okay, so notice that this is a little bit different from mutually exclusive. We have this little bit of extra on the end where we're actually subtracting out when it's possible for both of these events to happen at the same time. Okay, now that might be kind of confusing at first, but we'll go over a couple of examples where these events actually happen at the same time. Okay, a really good one is this first one. Okay, a card is drawn from a deck of 52. Okay, so when we talk about cards, deck of 52, these are your playing cards, your spades and clubs and diamonds and hearts. Okay, 13 cards in a suit. You're at your ace in two and three and four and five and six and seven and eight and nine and ten, and then your jack, your queen and your king, those kind of cards. Okay, all right, what we want to do from a deck of 52, find the probability of drawing a king or a heart. Okay, so we're only drawing one card. Find the probability of drawing a king, okay, or drawing a heart. If I can actually draw a good heart there. There we go, that's a pretty good one. All right, so now what makes this an inclusive event? Okay, inclusive means that basically there's a little bit of overlap when we talk about these two categories. Okay, so kings and hearts. Now there is one card that overlaps between these two categories. That is the king of hearts. Okay, so the king of hearts is actually one of the cards that occurs in both of these categories. Okay, the king of hearts is a king and it is a heart. Okay, so what that does is that creates a little bit of overlap between the two probabilities I'm going to have because if you look back up here, with my notation, these two probabilities I've got to kind of separate them out. But then what I'm going to do is I'm going to subtract the probability of both of these events happening. Okay, both of these events happen when I draw a king of hearts. Okay, so let's go through the rest of these examples. So now that I know it's an inclusive event because there is some crossover, I'm going to look at this as the probability of drawing a king plus the probability of drawing a heart minus the probability of drawing a king that is also and king and a heart. I can't really draw my hearts very well. Try that one more time. There we go. There's a good heart. Okay, so from here the probability of drawing a king, well there are four kings out of a total, that's a bad four. Try that one more time. Okay, there are four kings out of a total of 52 playing cards. Okay, over here we have the probability of drawing a heart. Now there are 13 hearts out of a total of 52 playing cards. But now here's the thing, the king is in this stack right here and the king is also in this stack over here with this 13. Okay, so the thing is we can't count this king of hearts, we can't count him twice. So that's why we have to subtract off any time that both of these events occur. Both of these events occur once with the king of hearts out of 52 cards. So I have to subtract one of them out. Okay, that's why we have this little bit of extra on the end here. Okay, so I can't count him, I can only count him once. Okay, so I count him once here, I count him once here but I can't count him twice, right? I can't have him in both spots. So I subtract one of them, doesn't matter which one, I just subtract one of them out. Okay, so this is going to be 4 plus 13 which is 17. 17 minus 1 is 16. 16 over 52. Reduce that fraction, I think both are divisible by 4. Yeah, I think both are divisible by 4. 4 goes to 52, 1, 3 times, 13 times. Okay, so the probability of drawing a king or a heart is 413. It's 4 out of every 13 times. All right. All right, now for another example, change my colors here a little bit, another example, find the probability of rolling an odd number or rolling a number greater than 2. So what we're talking about is the number cube here. It doesn't really explain exactly what we're doing. But again, if we're talking about odd numbers and numbers greater than 2 with rolling, okay, we're talking about number cubes. Okay, so find the probability of rolling an odd number or rolling a number greater than 2. Okay, notice I used n for number greater than 2. All right, so now these are going to be, this is going to be an inclusive event because when you roll a number cube, odd numbers are 1, 3, and 5. So odd numbers are 1, 3, 5, numbers that are greater than 2 are 3, 4, 5, and 6. Notice there's a couple of numbers I said twice. 3 and 5 are both numbers that are odd and that are greater than 2. So again, there is your little bit of overlap. There's your little bit of crossover between these two categories. That's what makes this an inclusive event. All right, so now that I know it's an inclusive event, I use the probability of me finding an odd number plus the probability of me finding a, or finding rolling I should say, a number greater than 2. But then I subtract out the probability of both of them happening that would be an odd number and a number greater than 2. Okay, subtract out all those extras. So the probability of rolling an odd number, that would be a 1, a 3, a 5. So that would be 3 out of 6 plus the probability of rolling a number greater than 2. That would be a 3, a 4, a 5, a 6. So that's 4 out of 6 possible rolls. Minus, now notice, actually stop real quick. Notice here, 3, 6 plus 4, 6. That would be 7, 6, which is a number that's greater than 1, which means it would be above 100% possibility that this could happen. That right there throws off some red flags, so you know that you're going to have to subtract something here. Okay, so the probability of an odd number and a number that's greater than 2. So your odd numbers and numbers greater than 2, that would be your 3s and your 5s. Okay, so that's 2 such events out of 6. So that's going to reduce us to a total of 5 sixth. Okay, so there is our probability. 5 out of every 6 rolls is going to get us either an odd number or a number that's greater than 2. Alrighty, and that's it. That's all I have for inclusive events. Now remember, inclusive events are the events that have a little bit of crossover. Like, for example, if I'm looking for the probability of a king or a heart, the king of hearts would be that crossover. Okay, if I'm looking for the probability of rolling a number cube and getting an odd number or a number greater than 2, the 3 and the 5 were the two numbers that are in both of those categories. So there's your little bit of crossover for those two. Okay. Alright, that was inclusive events. Thank you so much for watching. Hope you enjoyed the video and we'll see you next time.