Lesson 1: Sample Spaces & Events

In life, and certainly in trading and finance, we have to make decisions without knowing for a fact what's going to happen. Life isn't deterministic. Not everything is if you do A, B will happen. Sometimes we have to model uncertainty and make optimal decisions under that uncertainty, not knowing for sure what's going to happen. And probability and probabilistic modeling lets us do that. Probability gives us a way to represent and reason about the likelihood of uncertain events happening. If we assign a probability of .7 to a stock going up in a given day, that means there's a 70% chance of it going up. It's not guaranteed, but it means that it's more likely to go up than not. That was a probability .7 that we assigned to an event, which was the market going up. So, now we have two of the three building blocks of probability. So, let's talk about the sample space. The sample space is the space of every event that could happen. For example, if you flip a coin one time, the sample space is heads, tails. One of those two things will happen. In other words, the probability of something in the sample space happening is one. That's a certainty. From within the sample space, we can construct events. So, for example, let's define an event A that equals just heads. A is a subset of the sample space. So, what's the probability of A? What is the probability of flipping heads when we flip our coin? We know that that's 1/2 or 50% obviously. Or in other words, since we know that these events are equally likely, we can take the size of event A divided by the size of the sample space. One event, heads, out of the two possible events is our event that we defined, A. Okay, now let's say we're rolling a six-sided die. So, the sample space here is we roll a one, or we roll a two, or a three, or a four, five, or a six. Okay, now let's define a different event. Let's have this event be that we roll an even number. So, this includes two, four, or six. What is the probability of this event A? Again, all of the outcomes in the sample space have the same probability, so we can just look at the size of A divided by the size of the sample space. So, there are three events in A, so three out of the six total rolls that we could get constitute our event A. Okay, now we need to start viewing events as sets. So, in our same problem where we're rolling one die, let's define an event A, which is that we roll an odd number. So, that's one, three, or five. Now, let's define a second event B, which is that we roll a prime number. Within our sample space, the space of all the things that could happen, we have two events that we've defined, A and B, which are equally likely. B has a three in six chance of happening, and A also has a three in six chance of happening. But, they also overlap. They share some outcomes. If we roll a three, both event A and event B have happened. That