 All right, so another important rule for calculating basic probabilities that we'll make use of is the rule describing complementary events. So what I mean by this term complementary events, two events are complementary. If between them, they cover all the different possibilities. So as an example, if I flip a coin, either it comes up heads or it comes up tails, but those are the only two options. There's heads and there's tails and there's no other possibility. As a more chemistry related example, an electron can have spin up or it can have spin down, but those are the only two possibilities, heads and tails for a coin, or spin up, spin down for an electron are complementary events, because if we know one of them happens, the other one is the only other option. So the probability rule, if we have two complementary events, and again writing this as an equation, is just putting something that's fairly clear into an equation that might make it seem more difficult at first. But if I want to know the probability of one event A, it's just one minus the probability of the other event if A and B are complementary. So to make sure we understand how that works, we can think of this example of the butane molecule. So if I have a butane molecule that at room temperature in the gas phase is either anti-Gosch plus or Gosch minus, with probabilities of 68% in the anti-phase and anti-confirmation and 16% in either of the two Gosch confirmations. If I want to know what the probability is that the molecule is in a Gosch, either one of the two Gosch confirmations, either Gosch plus or Gosch minus, when we talked about mutually exclusive events, we thought of that as an or problem when we summed these two results. There's another way to do that, which is to say that either the molecule is in a Gosch confirmation or it's in an anti-confirmation and between those two, Gosch or anti, they cover all the different possibilities. So those are complementary events and we can say that the probability that it's Gosch is one minus the probability that it's anti or one minus 68.68 gives us 32 or 32%. Whichever way we think about it, either as mutually exclusive events or as complementary events, we get the same answer, of course, and this illustrates that often there's more than one way to think about a probability problem. Often what happens is you can think of one way that's a little bit easier to do the problem than another to show you an example of that. Let's consider a hydrogen atom. So again, I'll give you the probabilities, the experimentally determined probabilities for finding the hydrogen atom in different states, like we did for butane, but there's more than one state that the electron in a hydrogen atom can be. If I ask you what orbital you'd find the electron in a hydrogen atom in, you'd tell me it's in the 1s orbital. But if I raise the temperature of the hydrogen atom significantly, let's say we are talking about the hydrogen that's in the sun rather than in the laboratory here on earth, then maybe there's a possibility that electron could be in the 2s or the 2p or some higher level excited state. At a particular elevated temperature, the exact temperature is not terribly relevant for this example, let's suppose we've heated the hydrogen atom to a point where there's only a 78% chance the electron is found in the 1s orbital. And in fact, there's a 2% chance of finding it in the 2s orbital and a 6% chance of finding it in the 2p orbital. And I can keep going. The more higher excited states, 3s, 3s, 3p, 3d have their own individual probabilities. And I don't have to stop there. There's 4s and 4f and 5s and there's actually an infinite number of states if I continue this list. So those probabilities, eventually, you're going to have to add up to 100%, but I'm going to have to continue further down this list to get them to add all the way up to 100%. Let's suppose what I want to know is what's the total probability that that molecule, that atom of hydrogen is in some excited electronic state, that the electron is not in the ground state, the 1s state, but it's in one of the various excited states. So if I think about that as a mutual exclusive problem, then what I've asked to do is what's the sum of the probabilities in the 2s or 2p or 3s, 3p, 3d, and so on. So I'd have to add up all these numbers. But of course, the easier way to answer that question is to say, well, that's either it's in a ground state, the one single ground state, or it's in one of the excited states. So the probability that it's in any one of the excited state is 1 minus the probability that it's in the ground state. And that we can do much more easily. So there's 1 minus a 78% chance or a 22% chance that the molecule is in one of these excited states. So all these excited states are going to sum up to 22% because I know the whole list has to add up to 100%. So here's a case where it's clearly easier to think about this problem in terms of complementary events than by summing a list, which I may not even have the full list of things I need to sum up over here. So this illustrates that often thinking about the probability problem from a complementary events point of view gives you a shortcut versus calculating the answer the most direct way.