 So we need to introduce several basic ideas for computing probabilities, and the first of these is something we'll call mutually exclusive events, how to calculate the probability for mutually exclusive events. So it's actually much easier than it sounds. So as a simple example, we can say let's roll a die, roll a six-sided die. So we can get results of one, two, three, four, five or six. Say what we're interested in knowing is what's the probability that we either roll a five or we roll a six for that six-sided die. So that question may seem completely obvious. There's six things we can roll, two of them are five or six, so that the odds of getting a five or six is two different options out of the six or one chance out of three of getting a five or six. So like I said, easy problem to start out with, but when we explain the logic behind that answer a little more clearly, it'll help us introduce some terminology. So there's six different outcomes that we could have gotten. We could roll a one, we could roll a two, a three, a four, a five or a six. Any one of those outcomes could have happened when we rolled the die. Each one happens with an equal probability, and we've asked what's the probability that we get one of these last two outcomes. So each one of these, there's six outcomes, so there's a one chance and six of getting each one of these outcomes. So what we did intuitively without thinking about it much was we said chances were two chances out of six of getting our five or six, that's because there's a one chance and six of rolling a five and a one chance and six of rolling a six. So what we intuitively did without thinking about it was we used this rule, which describes what happens when we have mutually exclusive events. The probability of some event A happening or some event B happening, rolling a five or rolling a six is the probability of the first one plus the probability of the second one. And that rule happens as long as events A and B are what we call mutually exclusive, meaning if one of them happens, it excludes the possibility of the other one happening. If I roll a five, I can't also have rolled a six. If I roll a six, I can't also have rolled a five. Those two events exclude each other mutually. They're mutually exclusive. So introducing this terminology mutually exclusive and introducing an equation seems like it's made things a little more complicated than they needed to be for something we could have just answered without any equations. But when we move on and study problems that are more about chemistry and less about simple things like cards and dice, then it's useful to have these equations and terminology to fall back on. Just to do one example of something that's more like a chemistry problem rather than a dice problem. And it will be equivalently easy as the dice problem. Let's say we have a butane molecule, a molecule of butane C4H10. And butane can exist in three different confirmations. It can exist in the trans or anti-confirmation or a gauche confirmation. And the gauche confirmation can either be gauche plus or gauche minus. So those are the three different confirmations that a butane molecule can have. If you don't remember what the confirmations are butane are, that's not terribly relevant for this problem. But I'll tell you that at room temperature in the gas phase, butane, there's a 68% probability, 68% chance that if you grab a butane molecule out of the gas phase, it'll be in the anti-configuration, a 16% chance that it'll be in the gauche plus configuration confirmation and a 16% chance that it's in the gauche minus confirmation. We can just treat those as experimental numbers. We'll learn later on in the course how we can calculate numbers like that. But those probabilities add up to 100%. And let's say the question we want to know is overall what's the probability that a butane molecule selected randomly will be in one of the two gauche configurations, i.e. probability that it's in the gauche plus or the gauche minus configuration. So this is that same type of problem. If it's in the gauche plus configuration, it can't also be in the gauche minus. And vice versa, those probabilities exclude each other. They're mutually exclusive. So the way we can find out the probability that's either this or this is by adding those two things together, probability of gauche plus and probability of gauche minus combined. So 16% and 16% gives us 32%. So mathematically just as easy as the example with rolling a die, but one step slightly more confusing because it's talking about chemistry instead of talking about simple numbers like rolling dice. The key thing to remember is when you have an or problem, probability of one thing or another thing, if the two options are mutually exclusive, you just add the two individual probabilities together. It's important to remember to check for whether the options are mutually exclusive. And that won't always be the case. So as a third example, one where it's useful to remember this terminology and double check that things are mutually exclusive, we'll do an example with cards instead of with dice. Let's say out of a normal deck of playing cards, I draw one card. And I want to know what's the probability that I've drawn a queen or that I've drawn a heart, a card with a suit of hearts out of that deck of cards. So that sounds like an or problem. It sounds like I might want to say add the probability of drawing a queen to the probability of drawing a heart. The probability of drawing a queen out of the 52 cards in the deck, there's four cards that are queens and there's 13 cards that are hearts. So there's 13 chances out of 52 that I could draw a card that's a heart. So if I add those together, I would get 17 out of 52. But we know that's not the right answer. Those 13 cards that are hearts include the queen of hearts. There's only three additional queens that are not hearts. So adding these two numbers together doesn't give us the right answer. This is not the right answer. The probability of a queen or a heart is not the probability of a queen plus the probability of a heart. The right answer is only 16 out of 52. There's 16 cards that are either a queen or a heart. And the reason we didn't get the right answer with this equation is queens and hearts are not mutually exclusive. If I draw a queen, I could also have drawn a heart. Those two possibilities don't exclude one another. So that example is here just to show us that the simple rule is whenever you see an or problem, add the two individual probabilities. That works a lot of the time, but it only works if the events you're talking about are mutually exclusive. So our next step will be to understand the other basic rules of probability and then to see what happens when we combine them with one another.