I remember those fun middle school days when my math teacher used to teach us probability by bringing in a bunch of marbles and bags to find out what is the chance you pick a blue or red or pink marble, etc. Of course, we as college/high school students are gonna learn more than that. Probability is basically the 'science' and 'math' behind the uncertainty or certainty of an event. And it is applied EVERYWHERE. defining your sample spaceWhenever you are dealing with probability problems, you always need to define your sample space. The sample space is essentially the 'space' where you describe all of the possible outcomes of the event. It is also known as the symbol omega. find the probability lawThe next step is to find your probability law. The probability law is basically the 'formula' used to assign probabilities to events of an experiment. For example, in some experiments (which we will talk about soon), the probabilities are determined by the area the events take up in the sample space. This is called a continuous probability model. In other cases, the probabilities of an event are determined by simple counting. For example, lets say this was your sample space: The orange part in the diagram is called an event. An event is just a collection of outcomes in a sample space. Lets say omega (sample space) consists of 6 equally likely elements. What is the probability you randomly choose a dot in the orange part of the board? Well, we know there are 6 different possible outcomes, because there are 6 different points. So, we basically divide the probability of the event with the total number of possible outcomes. In that case, the probability will be 3/6. So a half. Not that tricky, right? discrete uniform lawSo the example we just went over supports the discrete uniform law. The discrete uniform law says that if the sample space consists of n equally likely elements, and A, which is a subset of the sample space, consists of k elements, Probability of A = P(A) = k * 1/n = k/n Since all of the elements are equally likely to be chosen, each element has a probability of 1/n. Since A has k elements, the probability of A being chosen is k * 1/n = k/n. intro to axiomsOk, so far, so good. But what if the probability of something is negative? Is that even possible? How do we put rules and limits to these probability values? This is where axioms come in. Axioms are certain properties that probabilities need to follow in order to be valid. These are the three main probability axioms:
The first axiom is the non-negativity axiom. This axiom tells us that all probabilities have to be greater than or equal to 0. They CANNOT be negative. A probability of 0 indicates it is uncertain to happen, whereas a probability of 1 tells us that it is certain. The next axiom tells us that the probability of the sample space, or omega is 1. This makes sense because the sample space consists of all the possible outcomes. So it is absolutely certain of the event is going to happen, since it is in the sample space. The last axiom is famously known as the additivity axiom. The additivity axiom tells us that if there are two events that are disjoint (we will talk about this soon), then the probability of both the events together is the sum of the individual properties. I know that was a lot of info, so lets break it down: Look at the above diagram. Lets say we have two events in our sample space, Event A (highlighted in yellow), and Event B (highlighted in red). The intersection of these two events is the orange part. You denote the intersection of two events like this: A ∩ B. The intersection of two events means that there is a chance A and B can happen at the SAME TIME. The union of two events means there is a chance A or B can happen. You denote the union of two events like this: A ∪ B. So in the above example these two events were NOT disjoint. Disjoint events are two events or more where no intersection occurs. When both the events don't have an intersection they are disjoint. So, In this case, Events A and B are disjoint. So, back to the additivity axiom. The additivity axiom says that, if the intersection of two events is 0 (disjoint), then the P(A ∪ B) = P(A) + P(B). So this is saying that the union of these two events is the sum of the individual probabilities. Which makes sense because they are both separate events and we want to find whats the probability we land in event A or B.
Anyway, probability is a tricky but really cool concept to learn. Good luck!
0 Comments
|
Archives
May 2021
Topics
All
|