 So, being able to calculate the multiplicity for some number of permutations or combinations is useful if, primarily if, each one of those different arrangements of some objects is equally likely as all the others, and then that allows us to calculate the probability of some permutation or some combination of those objects. But one way that molecules and chemistry problems are very unlike dice and coins is that different outcomes are typically not exactly the same probability as each other. Every outcome on a roll of a die is exactly the same, but different conformations of a molecule or different energy levels in a molecule don't all have the same probability of occurring. So we need to be able to understand what happens when the probabilities of each individual event are different from one another. So to consider that, we'll need to talk about this binomial probability distribution. But let's start by considering the case we do understand fairly well at this point, which is a case involving coins. So let's say we have, we flip ten coins, come up either heads or tails in each of ten flips, and let's say we want to know what's the probability that we get five heads and five tails. So just as an example, so that's the real question we want to know the answer to. As a warm-up, let's say I'll just write down some particular combination of heads and tails, one, two, three, four heads, one, two, three, four tails. So let's say here's a particular arrangement of five heads and five tails. What's the probability that if I were to flip ten coins, I would get a heads and then a tails and then a tails and then a tails and so on and exactly this pattern? Notice that that's an and problem, heads and tails and tails and tails and heads in that pattern. Those events are independent from one another. So I can say that that probability is the probability of heads times the probability of tails, but that's exactly the same as the probability of heads. So it's one-half times one-half times one-half. Every one of these outcomes I had a 50-50 chance of seeing the exact thing that I got since I flipped the coin ten times, that's one-half to the tenth. So that particular outcome has a probability of one over two to the tenth. That's not the same as asking what's the probability that I get five heads in any order and five tails in any order. If I go back to that question, probability of getting exactly five heads is the probability of getting any one of those events multiplied by the multiplicity. How many ways are there of getting five heads and five tails? So that's where the binomial coefficient comes into play. So each one of the events has probability one over two raised to the tenth. The multiplicity of getting five heads in ten coin flips is ten choose five. How many ways are there of choosing which five places to put the heads in in this list of ten outcomes? So as a number, I won't write out the factorials, but ten factorial divided by five factorial divided by five factorial. The number is 252. Two raised to the tenth power in the denominator, that number is 1024. So roughly 25% of the time will get five heads and five tails. So maybe pause for a minute and think about that. If I had asked you ahead of time if I flipped ten coins, what's the most likely outcome? Is it going to be five heads and five tails or one head and nine tails? You're probably going to think five heads and five tails is most likely because that's 50-50. And it is the most likely of all the individual outcomes, but only 25% of the time do I get exactly five heads and five tails. Getting four heads or six heads is not too unlikely either. But so that's how to do this problem. If we know that each one of these outcomes is exactly as likely as the other, heads is as likely as tails. Let's think about the more interesting problem now. I suppose our coin isn't fair. I suppose it's not a 50-50, whether it's heads or tails. So it's a different problem. What's the problem? The probability of getting a heads is 60%. And the probability of getting tails is the remaining 40%. So I have an unfair coin. So that's not super interesting when we're talking about coins. But that does begin to become interesting when we start talking about molecules. Because that begins to allow us to ask questions like, what if I have a molecule that has a 60% chance of being cis and a 20% chance of being in a trans confirmation? So thinking about it in terms of coins, however, we can say, what's the probability that out of 10 coin flips, I get five heads and five tails with this unfair coin? I think about the problem in the exact same way. It's the probability of any individual outcome. Let's take this outcome, for example, the probability of heads, tails, tails, heads, tails, heads, heads, tails, heads. But now instead of one half for each outcome, I've either got a 60% or a 40%. So it's still the probability of any one of these outcomes multiplied by the multiplicity, the number of ways that outcome can occur. The probability of one of these outcomes, let's think about this one. So probability of a head, that's 60%. I want to multiply that by the probability of tails. That's 40%. And then tails, I'm going to have another 40%, another 40%, a 60%, a 40%. When I multiply all the heads together, I'll have 0.6 showing up a total of five times. I'll have 0.4 showing up a total of five times. So in the end, I'm going to have 0.6 and 0.4 each multiplied by each other five times. That's the probability of getting this particular arrangement of heads and tails. But notice, it doesn't matter what order I multiply those together in. That's the same as if the heads and tails were in some other arrangement. As long as there's five heads and five tails, this is going to be the probability of that particular outcome. And I still have to multiply that by the ways of the number of arrangements, the multiplicity, ten choose five. So what's changed in this version of the problem, from this version of the problem, is not how many ways I can distribute five heads among the ten outcomes, but in how likely each one of those outcomes is. The heads have become more likely, the tails have become less likely, overall, the overall probability has changed. And if I just skip the math and write down what the calculator tells us, 0.6 to the fifth times 0.4 to the fifth times ten factorial over five factorial over five factorial, that works out to be 0.201 roughly. So there's only about a 20% chance of getting half heads and half tails for my unfair coin. It's a little less likely than it was before because the coin is no longer a 50-50 coin. But the important thing is now we can do a calculation like this, regardless of what the probability is of each individual outcome. In the general case, this binomial probability distribution that we're talking about, if I want to know, let's say I have event number A has some probability of occurring, some different event B has probability, everything else. So 60, 40, 50, 50 doesn't matter, but those two numbers have to add up to one. The interesting question that we can now write an answer to is, if I try something n times, what's the probability that little n of them come up with the result A? Then that's just, we've worked two examples of this so far, that's going to be the probability of one outcome times the multiplicity. Each one of the individual outcomes, I have to get A little n times, by definition, that's what I'm asking about. Each time I get A, I had a probability P, I get the other option B, that happens the remainder of the times, big n minus little n, and the probability of that happening each time was one minus P. So these things together are like the 0.6 to the 5, 0.4 to the 5, that's the probability of any individual outcome, and I still need to multiply that by the number of ways of getting little n out of big n options. So n choose little n. So this probability, probability of seeing outcome A little n times, is just the probability of it happening once raised to the number of times it happened, probability of it not happening raised to the number of times it didn't happen, times this multiplicity. So that's what we call the binomial probability distribution. Notice, be careful with the terminology here. Remember that this thing, big n choose little n, we call that a binomial coefficient. This probability, P to the n, 1 minus P to the n minus n times a binomial coefficient, that thing is called a binomial probability distribution. So there's some possibility of getting confused between those two terms, but one is talking about a probability, one is talking about a multiplicity, a count of the number of ways something can happen, keep those two terms in mind. As our last example, we'll do an actual real bona fide chemistry example now. Let's go back to our butane example, which we know has not trans, but anti, if we're using proper terminology. So a butane molecule has a 68% chance of being in the anti confirmation. And then 0.32 is the probability of being in either one of the two Gauch confirmations. So let's ask the question, let's say I have 20 butane molecules. And I want to know what is the probability that out of those 20 molecules, six of them are in the Gauch configuration. So collection of 20 butane molecules, I want to know, I might have all 20 in the Gauch confirmation, I might have none of them. I want to know what's the probability that six of them in the Gauch confirmation. That's exactly the type of problem we use this equation for. Probably that six of them are in the Gauch configuration out of 20 different molecules. That's going to be probability that one of them is Gauch, 0.32, raised to the sixth power. It's how many of them I want to be in the Gauch confirmation. Probably that they're not in the Gauch confirmation, 0.68, raised to the 14th power. That's the number that are not in the Gauch confirmation. And then I multiply that by the multiplicity, 20 choose 6. So we clearly need a calculator to solve that. But if we plug the numbers into the calculator, 0.188 or about 19% of the time, we find that our collection of 20 butane molecules will have six of them that are Gauch. So that's beginning to sound like problems that we couldn't possibly have tackled without some understanding of probability. And it's also beginning to show why these are relevant chemistry. Probability questions are relevant for answering questions about chemistry. The one thing that might occur to you to wonder at this point is, what if we have more than just two confirmations? If I don't just have anti and Gauch, but if I want to consider Gauch plus or Gauch minus or anti, or if I have a molecule with lots of different energy levels or lots of different conformational states, then using the binomial probability distribution isn't going to be enough. So the next video lecture will help us begin to tackle what to do if we have more than two options to consider.