Statistics the study of uncertainty. How do we measure it? And how do we make the decisions in the presence of it? One of the ways to deal with uncertainty, in a more quantified way, is to think about probabilities. Let's think about some examples of probabilities. Suppose we're rolling a fair six sided die and we want to ask what's the probability that the die shows a four? We could ask other questions, for example, is this a fair die? Does it make sense to ask, what is the probability that the die is fair? We could ask questions such as, what's the probability that it rains tomorrow? If we're dealing with the Internet, we may have a router that's passing through information. We can ask, what's the probability that it drops a packet? We might be comparing routers from two different companies. And we can ask, what's the probability that a router from one company is more reliable than a router from another company? We can ask more existential questions such as, what's the probability that the universe goes on expanding forever? There are three different frameworks under which we can define probabilities. We'll talk about all of them briefly here. The first one is the Classical framework. The second, there's a Frequentist framework, and the third one is a Bayesian framework. Under the Classical framework, outcomes that are equally likely have equal probabilities. So in the case of rolling a fair die, there are six possible outcomes, they're all equally likely. So the probability of rolling a four, on a fair six sided die, is just one in six. We could ask a related question, which is what's the probability of getting a sum of four on a pair of rolls. If we roll two dice, two fair six sided dice. Then what's the probability their sum shows a four? Well if we think about this, how many equally likely outcomes are possible on a pair of dice? There's six equally likely outcomes on the first die. There's six equally likely outcomes on the second die. So there are a total of 6 times 6, or 36 possible equally likely outcomes on the pair. Of those outcomes, how many will have a sum of four? We could roll a one, on the first die and a three on the second, a two on the first and two on a second, or a three on the first and one on the second. So there are a total of 3 possible outcomes out of 36 equally likely outcomes, and so that's a probability of 1 in 12. This Classical approach works really well and we have equally likely outcomes or well-defined equally likely outcomes. That's very difficult to apply in any of these other cases. We can then move on, to a frequentist definition. Frequentist definition, requires us to have a hypothetical infinite sequence of events, and then we look at the relevant frequency, in that hypothetical infinite sequence. In the case of rolling a die, a fair six sided die, we can think about rolling the die an infinite number of times, this is pretty straight forward. If it's a fair die, if you roll infinite number of times then one sixth of the time, we'll get a four, showing up. And so we can continue to define the probability of rolling four in a six sided die as one in six. This also applies to situations such as internet traffic going through a router. If we lose 1 in 10,000 packets, then we can define the probability as 1 in 10,000. This approach works great when we can define a hypothetical infinite sequence. But then we can ask other questions, and they become more complicated under this approach. For example, what is the probability that it rains tomorrow? In this case, we need to think about hypothetical infinite sequence of tomorrow, and see what fraction of these infinite possible tomorrows have rain, which is a bit strange to think about. Or, in the case of asking is this a fair dye? Well, if we have a particular physical dye, and we're asking, is it a fair die, then we can roll it a lot of times, but that's not going to change whether or not it's a fair die. And so either it is fair, or it isn't fair. All these rolls are not going to change that. And so under the frequentist paradigm, this probability is either 0 or 1. It's zero if it's not a fair die and it's one if it is a fair die. This is not exactly an intuitive answer. In the case of the universe expanding forever, we can ask, if this is a deterministic universe and the same thing happens, then again, the answer is going to be either zero or one because every time we play forward expansion of the universe, either it will expand forever or it won't. On the other hand, we might subscribe to something like a multiverse theory, where there are an infinite number of parallel universes that can exist. In that case, we can consider this infinite collection and ask what fraction of this infinite collection have universes that expand forever? Depending upon what we know about the universe, we might get different answers. The frequentist approach tries to be objective in how it defines probabilities. But as you can see, it can run into some deep philosophical issues. Sometimes the objectivity is just illusory. Sometimes we also get interpretations that are not particularly intuitive.