 Okay, so we've seen several rules that I've called the basic or elementary rules for calculating probability events. Unfortunately, real-world problems, chemistry problems, and other real-world problems are never as simple as calculating probabilities for simple things like dice or cards. So we need to know how to combine those rules to calculate probabilities for more complicated events or composite events that are not the simple case of just mutual exclusive or independent events or complementary events. So as an example, we'll stick with an example that involves dice rather than molecules to keep it relatively easy to think about, but the problem itself is going to be somewhat more complicated. So let's say we roll five dice and the question is going to be what's the probability? So we count up, when we roll the dice, we count up how many times we roll a six out of those five rolls of the dice and we're going to ask what's the probability that I get at least one six out of those five rolls of the dice. So this n that I'm asking for to be greater than one is the number of rolls of six in my five rolls of the dice. So there's several ways to think about that problem. I've rolled the dice five times. So if I want at least one six to show up, it could show up once, it could show up twice, three, four, five, up to a total of five times. So I could think about this problem as saying what's the probability that I rolled exactly one six or I rolled a six twice or I rolled a six three times or four times or five times. So far so good. I've broken the problem down. I've broken this composite problem down into a statement of a bunch of oars. So we recognize that as a mutually exclusive situation. Either I rolled it exactly one six or two six is three four five. Each of those possibilities excludes all the others. So I could just add calculate this probability and add it to this one, add it to this one, and add it to these two. And we'd certainly get the right answer if we did that. That's a valid approach to the problem. But each of those itself is a composite problem asking what's the probability that I rolled exactly three sixes out of five rolls. So the dice is going to ask, force me to rephrase the problem, break it down further and say did I roll the fives on the first and second and third roll or maybe the first and third and fifth rolls. So it's going to itself be a composite problem. The problem is a little bit easier if I think about the other probability rules. And I ask if I roll at least one roll of six, what's the composite? What's the other possibility? The only other possibility is I didn't roll any sixes. So either I rolled one or more sixes or I didn't roll any sixes. Those are the only two possibilities. So those two options are complementary. So the probability of rolling any sixes is one minus the probability of rolling no sixes at all. So this one it turns about is significantly easier to calculate. So the probability of rolling no sixes at all Let's think about what that means. I rolled the dice five times. I rolled one die five times or rolled five dice all together. But in order to roll no sixes I needed to get something that's not a six on die number one. And I needed to get something that's not a six on die number two. Each of the dice has to be something that's not a six. So notice that's an and problem. That's something that's not a six on die number one. And something that's not a six on die number two. So that's clearly an and problem. I need something to happen for die number one and die number two and die number three. Those are independent events. What happens on die number one is independent from what happens on die number two and so on. So I just need to calculate the product of these things. So because they're independent events probability of rolling something other than a six on die number one multiplied by the probability of rolling something other than a six on die number two. Let's go ahead and write that out. So probability of something that's not a six on die one times probability that's something that's not a six on die two and so on. I need to include five terms one for each of these five dice although I down to something that's not a six on die number five because I have five dice. So the full answer is going to be probably that I roll a six was one out of six. The probably that I don't roll a six is going to be five out of six. So I can either say that I just know what that answer is or if I wanted to I could break it down by saying rolling a six and rolling not a six are complementary events. So on die number one there's a five chance in six that I didn't roll a six. Die number two same thing. Die number three same thing. Die number four and die number five also same thing. So if I multiply all those together I need a calculator to do that but that turns out to be 3125 over 7776 that if we look back that's the probability that I didn't roll any sixes whatsoever. I wasn't interested in that. What I'm interested in is the probability that I rolled at least one six. So one minus that number so that's 4,651 out of 7,776 is the exact answer. If I want to think about that in terms of probability I can round it off a little bit. That's pretty close to 0.6. So there's a roughly a 60% chance that I'm going to roll at least one roll of a six when I roll five dice all together. So that's an arbitrary example unless we're playing Yahtzee maybe we don't care exactly what the odds are of rolling a six but here's a good example of how I can number one I can think of a problem two different ways one of which might be easier than the other. Number two it makes me use both the complementary events and the independent events rules in order to calculate the ultimate final answer or if I think about it the different way I could have calculated it using mutually exclusive events. So no matter how complicated the problem is we can break it down and get the final answer by using instances of these basic probability rules.