 All right, so if we're going to use probability tools to be able to estimate how likely things are or are not in a chemistry problem, then it turns out to be very important to be able to clearly define exactly what it is we're calculating the probability of. And this will introduce an idea that we will call microstates versus macrostates. So to explain what this is about, we'll start with a simpler example based on the lottery, for example. So let's say you have a friend who plays the lottery, and one night he watches the lottery numbers get drawn, and the lottery numbers that get drawn are five numbers, a 20, a 21, a 22, a 23, and a 24. And he says, that's amazing, can't believe the numbers came out like that. What are the odds? And you say, well, I've just been learning about probability, I can help you figure out what the odds are, tell me how the lottery works. And he says, well, there's 80 numbers in a bucket, and you draw five of the balls out of the bucket, and these are the numbers that came out. And they don't have to come out in this order, they can come out in any order they want, and this would be the same result. So you say, well, I know exactly how to calculate that. You can say, first of all, if there's 80 numbers and five of them got chosen, then there's 80 choose five different possible results that can come out of the lottery. So 80 factorial over 75 factorial, five factorial. That works out to be about 24 million. So the probability of that happening, one over 24 million, that's a number that turns out to be one in 24 million or so, or about four times ten to the minus eight. So you can say, you can watch the lottery 24 million times and you'd only see something like that happen once. That is genuinely incredible. So then the next day, he watches the lottery numbers get drawn again. And today, that day, the lottery numbers, let's say, are something that looks much more random. This particular collection of numbers, and he says, well, so what are the odds that that would happen? And you say, well, I can calculate that too. If you have 80 numbers and you draw five of them, the odds that those particular five numbers came out is 80 choose five, and the probability is going to be exactly the same thing as it was before. It's only one in 24 million. You can watch the lottery 24 million times, and you'd only see that collection of numbers once. And he says, that doesn't make any sense. The first one is clearly much more rare than the second one. You don't know what you're talking about. So what is the problem here? Why is it that we got the same probability of this unique looking string of lottery numbers and for this much more common looking string of lottery numbers? And the answer to that question is we haven't been careful enough when we define exactly what we're talking about calculating the probability of. So what we've calculated the probability of is drawing these exact five numbers. When your friend was more interested in is maybe what is the, perhaps, so what we calculated was the probability of some exact outcome. And he might have been interested in the question, let's say, let's do this first, let's say. He wasn't really interested in knowing what's the probability that the lottery numbers were going to be a seven and an 11 and a 27 and a 43 and a 61. He was probably more interested in the probability that the five lottery numbers are consecutive, which is what surprised him in the first place, or that the five lottery numbers are not consecutive, don't have any consecutive numbers involved. So that second set of questions, whether the numbers are consecutive or not, that's a particular category of outcomes. We're interested in what are the numbers are consecutive, all of them consecutive, some of them consecutive, none of them consecutive. That's a very different question than calculating the probability of this exact set of numbers coming out. So we can distinguish between those. We've seen that probably the exact outcome might be a very small number. If we calculate, for example, the probability of five consecutive numbers, probably your friend would have been equally amazed if the numbers had been 30, 31, 32, 33, 34, in addition to 20 through 24, or even 21 through 25 instead of 20 through 24. So any string of five consecutive numbers would have struck him as very unusual. So there the odds would have been, the odds of any one of those outcomes multiplied by the number of different consecutive outcomes we can have. So the first number could have been anywhere from a one through a 76, which would result in a final number of 80. So multiply that by 76. And what do we get when we do that? So that's somewhat more likely, still pretty rare, 3.2 times 10 to the minus 6. About one time in 300,000, the lottery numbers will come out as five consecutive numbers. As a different category of outcome, we could say, what's the probability that there's no consecutive numbers in the lottery draw? So that's a somewhat more challenging probability calculation to work out. We won't do it here. It certainly can be done with a combination of the various basic probability tools we've talked about so far. But if we set up the problem and ask ourselves, what is the probability that none of these numbers ended up consecutive or next to any of the others? That answer turns out to be 7,685, if you're particularly interested in the value. So about 76% of the time, 77% of the time, the numbers won't have any consecutive numbers in there whatsoever. So these are the numbers that match your friend's intuition when he said that number, this sequence of numbers is very unlikely and this one seems very normal. He's not asking about the particular outcome, but the category of outcomes of whether numbers are consecutive or not. So this is what we mean, but when I say we want to define between microstates and macrostates. So microstate, these are beginning to sound more like science words rather than probability or lottery related words. So microstate is intended to describe the microscopic arrangement of particles in a system. If we're particularly interested in the lottery example of exactly which numbers we got as one of the outcomes, then we're describing a microstate. But more often than that, we're interested in describing some category of outcomes and we'll describe that as a macrostate. We're not interested in the microscopic position of every particle in the system, but some macroscopic description of how the particles are behaving. So this is, and we'll have more examples of this in just a second. This is a little bit reminiscent of the coin flip problem we've talked about when we talked about the binomial distribution to give you some examples of microstates as opposed to macrostates. If I say what's the probability that I get a particular sequence of heads and tails, then what I've described is a microstate. I can calculate the probability of this exact combination of heads and tails microscopically describing what the outcome of every individual outcome every individual event was. Before I could ask a different question, what's the probability that I were to get five heads and five tails if I flipped a coin ten times? So that includes not just this microstate but a bunch of other microstates that all obey that macroscopic description, that larger scale description of how many heads and how many tails I've gotten. So we've seen before how to calculate those numbers in a more chemistry context. So that would be an example of a macrostate. As a chemistry example, if I were to say to describe a microstate, I might say what's the probability that molecule number one is at some position r1 with velocity v1 and energy e1 and molecule number two has its own position, velocity, energy, and so on. So I've microscopically described the properties of every individual molecule in the system. That would be an example of a microscopically described state, a microstate. What I mean when I talk about a macrostate is something more like we typically think of in chemistry problems if I say what is the probability that the whole system has some total energy and some total volume and some total temperature, some total collection of properties. Then I'm describing the combined properties of all the molecules in the system rather than the microscopic descriptions of individual particles. So that would be an example of describing a macrostate for the entire system. So it's important to be able to understand whether we're asking a question about microstates or whether we're asking a question about macrostates because that's going to drastically affect the type of answers we get for these problems.