 Let's move on to the question of finding theoretical probabilities. And so if we want to put a framework on this, remember that the frequentist viewpoint holds that the probability of an event is the relative frequency with which the event occurs when I repeat the experiment a very large number of times. And in certain cases this is reasonable to do. If I want to find the probability that a coin lands heads, I could sit here flipping the coin a couple hundred thousand times and see how often the coin lands heads and that will give me my empirical probability. On the other hand if we actually want probability to be useful it might be helpful to know the probability before we run the experiment over and over again. What's the probability a hurricane will hit Miami Beach? And it'd be kind of nice to know what this probability is before we allow a whole bunch of hurricanes that direction. So we want to find some way of finding the theoretical probability of an event. And again here's we're having that second viewpoint of probability is useful. If I think about this from a Bayesian viewpoint, the question that I want to answer is what's the confidence? What's our confidence that the event will occur the next time I run the random experiment? And there's a couple of ways we might do this. To begin with let's introduce a new idea which is that a sample space that consists of equally likely outcomes. And the definition for this is that the probability of any one of these outcomes in our sample space is the same as the probability of any other outcome. And that either means if you are a frequentist that the outcome will occur as often as anything else. Or if you are a Bayesian that means that your confidence that the outcome will occur is the same for that outcome as it is for any other outcome in the sample space. And this leads us to the all-important question. Given a sample space how can I tell if it consists of equally likely outcomes? Well there's an easy way of determining this if you're told that the sample space consists of equally likely outcomes. Well what if you're not told that the sample space consists of equally likely outcomes? You're asking for a lot there but unfortunately the problem is that there's no easy way to determine if a sample space consists of equally likely outcomes. Unless somebody tells you that it does and most importantly you are willing to believe them when they tell you that. There is no easy way to determine when a sample space consists of equally likely outcomes. We must rely on our experience with the real universe. And what that means ultimately is we want to look at this from a Bayesian viewpoint. So consider the following experiment. Find the first letter of a person's name. And so let's consider this question from a couple of different perspectives. First of all are we actually talking about something that is a random experiment? And if so let's take a look at the sample space and ask ourselves does it consist of equally likely outcomes? So let's see is it random? Well yes again it fits our definition for mathematical randomness because we cannot predict with certainty what the first letter of a person's name will be. So it qualifies as a random experiment. And so we might consider what's our sample space going to look like? Well if I run the experiment find the first letter of a person's name then the possible outcomes are the first possible letters. And these are the letters A through Z. So there's our sample space. Now consider this question do we have a set of equally likely outcomes? And so here are the possible outcomes and so the question is do we have a feeling that some of these outcomes are either more likely to occur or less likely to occur? And again if we want to look at this from a Bayesian framework we want to ask ourselves how confident are we that a particular letter appears? And so you might think about the names of all the people that you know and most likely you'll probably be more confident that T will be the first letter of a person's name than X will be. And the fact that we are more confident that T will be the first letter than X says that the probability that T will be an outcome is greater than the probability that X will be an outcome. And so these outcomes in this sample space are not equally likely. So from a Bayesian point of view that difference in confidence translates into a difference in probability and so our sample space is not going to consist of equally likely outcomes. Let's consider a different sample space. Suppose I flip the coin one time and let's say click it with the sample space and see whether it determines consist of equally likely outcomes. So the possible outcomes if I flip a coin either the coin lands heads or the coin lands tails. And I have two possible outcomes and the question is from my experience with the real universe do I have a feeling that one of these is more likely to occur than the other? And our experience with the real universe says that we kind of feel that these two are equally likely to occur. We don't feel confident that one of these is going to occur more often than the other. So we might conclude that the two outcomes are equally likely and our sample space consisting of these two outcomes is going to be a sample space of equally likely outcomes. Now here's an important observation to make which is that this notion of equally likely outcome seems to be very closely tied in the minds of non-mathematicians non-probableists to the idea of random. Most people would describe this as a random experiment because the outcome is not predictable in advance. On the other hand this experiment find the first letter of a person's name most people would not say is random and their reasoning would probably be along the lines of some letters are more common than others and so what they're actually using here as their stand-in for random is not equally likely and it's important to remember the definition of random has nothing to do with how likely or unlikely an event is it has everything to do with whether or not it is predictable with certainty. It is that uncertainty that makes the experiment random. Now the focus on equally likely outcomes does make it important to really carefully consider what our sample space looks like. Let's consider a single experiment but this time we'll describe the sample space in two different ways. So here we're going to flip a coin three times and let's see if we can describe two distinctly different sample spaces for this experiment. So one of my sample spaces might be well let's see I flip a coin three times so maybe I'll get zero heads or one head two heads or three heads in three flips. So there's one possible description of the sample space but it's also possible to describe the sample space by indicating the results on each of the three flips. So maybe my three flips are heads then heads then heads or maybe heads heads tails heads tails heads heads heads or maybe it's going to be heads tails tails tails tails tails tails tails heads so there's a second possible description of the sample space. The same experiment I'm still flipping a coin three times but I have two different ways of describing the sample space. Now for a variety of reasons it's actually really convenient to be able to work with sample spaces that consist of equally likely outcomes. So in general we know that most sample spaces do not consist of equally likely outcomes and if you ever come across a sample space that you believe does consist of equally likely outcomes see rule number one. So going back to our coin flip example consider this sample space zero heads one head two head and three heads and do we have a set of equally likely outcomes? Well from a Bayesian point of view our confidence in one of these outcomes is greater or lesser than the probabilities will be different and we won't have a set of equally likely outcomes. So consider this outcome zero heads in three flips. On some level our experience with the real universe says that this outcome is going to be less likely than getting say one head on three flips and it is that experience with the real universe that causes us to think that some of these outcomes are less likely than other outcomes and so we have a sample space that does not consist of equally likely outcome. On the other hand consider our other sample space again for the same experience that the coin three times and this time I'll take a look at this set of experiments outcomes. So does this consist of equally likely outcomes? This is actually a harder question to answer so we might approach it this way. So it's like take any two of these outcomes for example heads heads heads or tails tails heads. Now we have not yet talked about calculating any probabilities but we should be very very careful in how we proceed. The question is do we feel that one of these outcomes is more or less likely than the other outcome and so one way we might approach this is the following. Let's consider this outcome HHH. This is the outcome that we get heads then heads then heads and it seems that the probability of this outcome is going to rely on the probability of getting heads then heads then heads. Now we haven't committed ourselves to what the actual probabilities are going to be but let's suppose that that probability is P then the probability of getting this result heads heads heads is based on the probability of P three times. Now let's consider the other outcome tails tails heads. So the probability of this outcome seems to depend on the probability of getting tails then tails then heads. Now earlier we expressed a belief that heads and tails separately seem to be equally likely outcomes so it also seems that the probability of this outcome tails tails heads also seems to be based on the probability P three times once for tails once for tails once again for heads and so it seems that both of these outcomes heads heads heads tails tails heads both of these outcomes seem to be based on the probability P used three times. And And because they're based on the same probability, used the same number of times, it seems reasonable to conclude that these are equally likely outcomes. And because the argument can apply to any other outcome in our sample space, it seems that we do in fact have a sample space that consists of equally likely outcomes. And that is an important step in what comes next.