Lecture Note
University
Duke UniversityCourse
Data Science Math SkillsPages
3
Academic year
2022
KatrCrayon
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68
Probability Definitions and Notation Probability is a measure of the likelihood that an event will occur. It is numericallyexpressed as a number between 0 and 1, where 0 indicates impossibility, and 1 indicatescertainty.Probability can be understood as the relative frequency of occurrence of an event in alarge number of trials. If an experiment is repeated many times, and each time the result isnot 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 possibleoutcomes. This section covers the basic principles of probability theory, including definitions ofprobability 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 confidencein its accuracy. The probability value assigned to a statement indicates the degree ofuncertainty surrounding it. When I am certain that a statement is true, I assign it a probability of 1. If I am certain thatthe 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 toassign some probability to this statement, and then after learning the true state of theweather 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 astatement 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 isnot 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 isimportant to keep in mind that, even before all the facts about a situation are known, theprobabilities 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 thinkthat 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 adistribution must have a non-zero probability assigned to it. This law illustrates the basicprinciple of probability distributions—that all outcomes within the distribution must sum to1. A probability distribution is a collection of statements, two or more, that are exclusive andexhaustive. 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—nomore of one of the statements can be true. So it should be obvious that we have thestatements it is raining and it is not raining, only one of those statements can be true at atime. 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 representsmy uncertainty to the statements that it is raining and that it is not, but when myinformation 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 chooseone 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 theprobability that I might draw an ace. Suppose further that I want to know the probabilitythat 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 adeck is well-shuffled, we can use the principle of indifference to assign a probability of 1/52that 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 theevent 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 thereare 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 ourprobability of drawing a queen is 1 in 13. To calculate the probability of a six-sided die landing on even, we must determine thenumber 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 comeup 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 ofprobability problems.
Probability Definitions and Notation
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