Probability - Robert P. Dobrow

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      There are many compelling reasons for including simulation in a probability course. Simulation helps build invaluable intuition for how random phenomena behave. It will also give you a flexible platform to test how changes in assumptions and parameters can affect outcomes. And the exercise of translating theoretical models into usable simulation code (easy to do in R) will make the subject more concrete and hopefully easier to understand.

      And, most importantly, it is fun! Students enjoy the hands-on approach to the subject that simulation offers. It is thrilling to see some complex theoretical calculation “magically” verified by a simulation.

      To succeed in this subject, read carefully, work through the examples, and do as many problems as you can. But most of all, enjoy the ride!

      The results concerning fluctuations in coin tossing show that widely held beliefs …are fallacious. They are so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory predicts. The record of a simulated experiment is therefore included….

      —William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, Third Edition (1968), page xi.

      The beginning is the most important part of the work.

      —Plato

      Learning Outcomes

      1 Define basic probability and set theory terms.

      2 Give examples of sample spaces, events, and probability models.

      3 Apply properties of probability functions.

      4 Solve problems involving equally likely outcomes and using counting methods.

      5 (C) Explore simulation basics in R with a focus on reproducibility.

      Probability begins with some activity, process, or experiment whose outcome is uncertain. This can be as simple as throwing dice or as complicated as tomorrow's weather.

      Given such a “random experiment,” the set of all possible outcomes is called the sample space. We will use the Greek capital letter normal upper Omega (omega) to represent the sample space.

      Perhaps the quintessential random experiment is flipping a coin. Suppose a coin is tossed three times. Let H represent heads and T represent tails. The sample space is

normal upper Omega equals StartSet normal upper H normal upper H normal upper H comma normal upper H normal upper H normal upper T comma normal upper H normal upper T normal upper H comma normal upper H normal upper T normal upper T comma normal upper T normal upper H normal upper H comma normal upper T normal upper H normal upper T comma normal upper T normal upper T normal upper H comma normal upper T normal upper T normal upper T EndSet comma

      An event is a set of outcomes, and as such is a subset of the sample space normal upper Omega. Often, we refer to events by assigning them a capital letter near the beginning of the alphabet, such as event upper A. The event of getting all heads in three coin tosses can be written as

upper A equals left-brace Three heads right-brace equals left-brace upper H upper H upper H right-brace period

      Event upper A contains a single outcome, and clearly, upper A subset-of-or-equal-to normal upper Omega. More commonly, events include multiple outcomes. The event of getting at least two tails is

upper B equals left-brace At least two tails right-brace equals StartSet upper H upper T upper T comma upper T upper H upper T comma upper T upper T upper H comma upper T upper T upper T EndSet period

      We often desire probabilities of events. But before learning how to find these probabilities, we first learn to identify the sample space and relevant event for a given problem.

       Example 1.1 The weather forecast for tomorrow says rain. The number of umbrellas students bring to class can be considered an outcome of a random experiment. If at most each of students brings one umbrella, then the sample space is the set . The event that between 2 and 4 umbrellas are brought to class is

      Dice are often used to illustrate probability concepts. Unless stated otherwise, in this text, rolling a die refers to rolling a fair six-sided die with the usual numeric labels of the numbers 1 through 6.

       Example 1.2 Roll a pair of dice. Find the sample space and identify the event that the sum of the two dice is equal to 7.The random experiment is rolling two dice. Keeping track of the roll of each die gives the sample spaceThe event is The sample space can also be presented using an array format, where the rows denote the first roll and the columns denote the second roll. The cell entries are the sum of the row and column numbers. All 36 outcomes will be represented in the resulting cells. The event can then be identified by finding the cells that correspond to the desired criteria.

      

       Example 1.3 Yolanda and Zach are running for president of the student association. One thousand students will be voting, and each voter will pick one of the two candidates. We will eventually ask questions like, What is the probability that Yolanda wins the election over Zach by at least 100 votes? But before actually finding this probability, first identify (i) the sample space and (ii) the event that Yolanda beats Zach by at least 100 votes.(i) The outcome of the vote can be denoted as , where is the number of votes for Yolanda, and is the number of votes for Zach. Then the sample space of all voting outcomes is(ii) Let be the event that Yolanda beats Zach by at least 100 votes. The event consists of all outcomes in which , or . That is,

       Example 1.4 Diego will continue to flip a coin until heads appears. Identify the sample space and the event that it will take Diego at least three coin flips to get a head.The sample space is the set of all sequences of coin flips with one head preceded by some number of tails. That is,The desired event is . Note that in this case both the sample space and the event are infinite, meaning they contain an infinite number of outcomes.

      What does it mean to say that the probability that upper A occurs or the probability of upper A is equal to x?

      From a formal, purely mathematical point of view, a probability is a number between 0 and 1 that satisfies certain properties, which we will describe

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