Hello. Welcome back. Okay, so this is where we left off. We had simulated a dice game where we were rolling a pair of dice and some in the spots that landed face up in the game. We had several observations that we, right going through them. Basically, we're not able to predict with certainty what the next throw will. Sometimes you show the teapot that really shows how how that looks. We also made the observation in that same vein that we sitting in a particular throw. We couldn't look backwards. And you have any idea what the previous her name was? We also noticed that some combinations of, di spots occur with higher frequency than others. So we a table of the 500 dice throws, and it just kind of looked at it. We also see the instagram shown here, so we noticed that there's some sort of pattern that is emerging, but it's not completely informative as to how to make a prediction. So the pattern, if we were to take a whole bunch more throws, might be described as, triangular. Right. So at the edges two and 12 or less frequent, Leigh observes, than the middle round, the sum of, say, 78 six. So, roughly speaking, what we're doing is we are formulating belief. Okay, so we were to restate that if someone were to ask us, what do you believe the next throw will be? We might guess 67 or eight. if someone was to, turn that around and say, Well do you want to bet against the occurrence of any particular some, you would probably say two and 12. So the question now becomes, how do we go from this hunch to a full understanding of this game that might inform how we how we might bet on what the next threat would be. The answer is statistics and probability. To get there we need some terms. So that's what we're going to do. This video is talk through some terms that are necessary or at really formative for probability. So we're going to start here, So sample space outcomes and events. These are really just so first off outcomes outcomes are the exhaustive list of possible results in some sort of experience So in the dice game, where the outcome is a some it's the numbers two through 12 events, events are collections of outcomes, so events are really up to us to define what we are interested in. So, for instance, we might not be interested in the elementary outcomes two through 12 in our nice game. What we might be interested in instead is the some odd versus even so we might want to bet more like a It's a roulette wheel where we're betting red versus black sample space is really the set of all possible outcomes, and these things are kind of used in formulating probability spaces as well. Talk through well as another example. Let's say that our experiment for game is throwing three coins and accounting heads where we're concerned on whether or not we receive an even number of heads. So even states the outcome is going to be the count of heads of 012 or three performance requirements. So the events that we're interested in is essentially evens versus odds, so even being zero to all to be one or three sample space, of course, is the full set of outcomes. If we're being fully pedantic. We have to talk about event spaces and really, I'm going to just list the conditions and we're not going to go to into the space that. But the event space has to conform to a series of conditions listed here. The reason We're concerned about the event spaces because that's the space that gives us the results that were interested in track. So the event space has to contain sample space just here. The event spaces close under compliments. In other words, if the event is there, the compliment or not, event has to also be in that space, and the event space has to be closed under candidate. We're going to use this two create some probability states. So if we turn to probability now we have We can use what we just defined to define all of it. So let's start with So first off, we create a probability space. So we had a, an event space. Now we have an associated probability space, so the probability spaces combination of the sample space event space and a probability function so problem most of us are more familiar with talking about in terms of probabilities, the probability function. So probability function is a real valued function that maps events to the real number line between 01 inclusive. This probability function adheres to the so called axioms of probability of covid Carl vaccines, which are really based on the set theory of statements that we just talked about. Some if we summarize them, they are listed here. So the first is that the probability of an event is a real number in the interval. So probability that something happens is somewhere between zero. The probability of at least one event occurrences. Something has to occur with the probability of so we form. We perform our roll of the die. There's going to be an outcome with problem if we are looking at events that are mutually exclusive, so two events that cannot occur at the same time. And we're interested in what is the probability of one or more of those events we can simply some the probability of the individual events again, This has to be events that are mutually exclusive, so we're starting to get some idea of the math behind, dealing with probability. So again, a probability space involves the sample space as we talked about the event space and the things that we're interested and a probability function probability function is a mapping of occurrence of events to the real number line within the city. What we're going to turn to next is rules for dealing with probability.
