Probability Definitions and Notation

Probability Definitions and Notation Probability is a measure of the likelihood that an event will occur. It is numerically expressed as a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. Probability can be understood as the relative frequency of occurrence of an event in a large number of trials. If an experiment is repeated many times, and each time the result is not the same, then it can be said with confidence that if we perform this experiment again, it will yield different results. In mathematical terms, probability is defined as: The number of ways in which an event can occur divided by the total number of possible outcomes. This section covers the basic principles of probability theory, including definitions of probability and probability distributions. Probability is a mathematical way to assign a value to the truth or falsity of a statement, with 0 indicating no belief in the statement's truthfulness and 1 indicating total confidence in its accuracy. The probability value assigned to a statement indicates the degree of uncertainty surrounding it. When I am certain that a statement is true, I assign it a probability of 1. If I am certain that the statement is false then I assign it a probability of 0. So let's say I'm sitting in my office and I don't know whether it's raining or not. I'm going to assign some probability to this statement, and then after learning the true state of the weather outside, I'll either have certainty that this statement is true or not true. Thus, certainty is that the sentence is either true or false. P(x) - probability of x ∼x - negation of statement x The tilde is also used to indicate the negation of a statement, and when we have a statement and its negation, we have a simple binary probability distribution. Any time we have a statement like "It is raining," and the negation of that statement, "It is not raining," those statements together will form a probability distribution. If we have complete information about a situation, one of the statements must be true. It is important to keep in mind that, even before all the facts about a situation are known, the probabilities for each of those two statements must still add up to 1. P(x) + P(∼x) = 1 If I think there is a 3 out of 4 chance that it is raining outside right now, then I must think that there is a 1 in 4 chance or 25% chance that it is not raining outside right now. If P(x) = 1, then P(∼x) = 0, and vice versa In probability theory, the law of the excluded middle states that every outcome in a distribution must have a non-zero probability assigned to it. This law illustrates the basic principle of probability distributions—that all outcomes within the distribution must sum to 1. A probability distribution is a collection of statements, two or more, that are exclusive and exhaustive. X = {x1, x2, x3, ..., xn} P(x1) + P(x2) + P(x3) + ... + P(xn) = 1 Exclusive means that given complete information—that is, all possible states of nature—no more of one of the statements can be true. So it should be obvious that we have the statements it is raining and it is not raining, only one of those statements can be true at a time. In addition, statements comprising a probability distribution must be exhaustive. When we have complete information, at least one of the statements must be true. I am sitting inside, unsure whether it is raining or not. I assign a likelihood that represents my uncertainty to the statements that it is raining and that it is not, but when my information is complete, only one of those statements can be true. In many situations, we have more than two statements forming a probability distribution. And in many cases, we have a large number of statements and no real basis to choose one outcome as more probable than another. Suppose we have a deck of 52 cards, in that case n=52. And I am wondering what is the probability that I might draw an ace. Suppose further that I want to know the probability that I would draw a spade ace from a well-shuffled deck of 52 cards. As far as I know, there is nothing special about the ace of spades. But assuming that a deck is well-shuffled, we can use the principle of indifference to assign a probability of 1/52 that any given card is an ace. According to the principle of indifference, we can calculate many probabilities as follows: The probability of a certain event is defined as the number of outcomes that are in the event divided by the total number of possible outcomes in the universe. In our deck of cards example, we might say that our event is drawing a queen and there are four queens in the deck. So, we have four outcomes within the definition of the event: queen of hearts, queen of diamonds, queen of spades and queen of clubs. We have a total number of possible outcomes that is equal to the number of cards, so our probability of drawing a queen is 1 in 13. To calculate the probability of a six-sided die landing on even, we must determine the number of possible outcomes that meet the definition of even. There are two, four and six. There are three ways to roll an even number on a six-sided die and six ways for it to come up odd, so the probability that the die comes up even is three over six or one half. The concept of permutations and combinations allows us to solve a large number of probability problems.