 All right, we've seen with the binomial probability distribution that that's useful for calculating the probability of some number of events happening when there's only two options, maybe A and B, or heads and tails, or spin up and spin down. More commonly though, we have more than two choices, more than two outcomes or results that can happen for molecules. So we need to be able to handle more than just two events, so that's where we need the multinomial probability distribution. So suppose we have a total of big N events that are going to happen, let's say in addition these are independent events, they might be coin flips or rolls of the die, or they might be individual molecules that behave independently from one another. And those events can have some number of outcomes, for example heads and tails, or molecules can be in energy level one or two or three or four and so on. And those outcomes happen with different probabilities. Outcome one, whatever that is, might happen with probability P1, outcome two might happen with probability P2, and so on. We might have a very large number of these different outcomes. Then what we need to know is we'd like to understand what's the probability that I get some number of outcomes of type one, some number of type two, and so on. So these numbers are, and one is the number of outcomes that we see out of these grand total of capital N that have type one and two would be the number of type two and so on. So if those are binomial distribution, if we only have two possibilities then we already know how to write down the answer. That would be if we want the first event to happen, first type of outcome to happen N1 times. Each time it happens, it happens with probability P1, multiply that together N1 times because the events are independent. The second type of event happens with probability P2 and that happens N2 times, so we multiply those together as well. And then the number of ways of shuffling these independent N1 outcomes together that are all the same, we would multiply that by big N choose little N. So that's what the result would look like if it were a binomial distribution. That's not what we have here. We have more than just two outcomes. We have perhaps three or four or more outcomes. So we also need to include P3 and P4 and so on. And then instead of a binomial coefficient, we would have a multinomial coefficient with an N1, an N2, N3, dot, dot, dot. So that's the only difference between the binomial and the multinomial probability distribution is we include all the different outcomes when we multiply the probabilities together for each set of outcomes and then we use a multinomial coefficient instead of the binomial coefficient to calculate the multiplicity. So let's do an example and make sure that's clear. And we'll use our butane molecule for that example, where again we know that at room temperature we've used these numbers before. The probability of being in the anti-configuration is 68%. The probability of being in the Gauche plus configuration is 16%. The probability of being in the Gauche minus configuration is 16%. And then what we'd like to know is let's say what is the probability that with N equals 12 butane molecules independent from one another, what's the probability that there are, let's say, four anti, four Gauche plus and four Gauche minus. So equal number in each confirmation. So the result of that calculation is we just plug those numbers into the binomial probability distribution. So we need four molecules in the anti-configuration, 68% chance that any individual molecule is in the anti-configuration. 0.68 to the fourth is the probability that it's going to happen four times. 0.16 to the fourth is the probability that a Gauche plus event with probability 16% will happen four times. And then in addition to that, 0.16 raised to the fourth is the probability that a Gauche minus event will happen four times. And we need to multiply all of that by the binomial or the multinomial coefficient. 12 choose 444 on the bottom. So that would be a 12 factorial divided by 4 factorial, 4 factorial, 4 factorial. So this is a relatively small number. These fractions multiplied by themselves a number of times, multiplied by a relatively large number. When we combine all those two things together, first of all, the decimals by themselves give us 9.2 times 10 to the minus eighth. And the multinomial coefficient is 34,650. So again, the multinomial coefficient's large, the probabilities are relatively small. When we combine those two things, the net result is .0032. So the numerical answer to the question, what is the probability if I have 12 butane molecules under these conditions that an equal number of them are in each of the three different states? There's only about a .3% chance that that's going to happen. There's other possibilities that are slightly more likely, such as, for example, most of the molecules being in the anti-state, because we know that that happens most of the time. But regardless of which individual combination of occupation of these states we're interested in, we can calculate that now with the multinomial coefficient. So this completes the set of basic probability rules that we're going to need. And the next step will be to start using those to calculate properties like these and more interesting ones for real chemical systems.