**Probability and Chance**

# 1. Introduction

Characteristic of this definition of probability can be seen in the experiment of flipping a coin. When a fair coin is flipped, there is an equally possible chance of the coin landing on either heads or tails. Since there are only two equally possible outcomes of this event, the probability that the coin will land on heads (an event) is 1/2. The same can be said of almost any game of chance. Consider rolling a die, the probability of rolling a specific number would be 1/6 as there is only one favorable case of the die landing on that number and six equally possible cases of the die landing on each number. The concept of equally possible cases will be a recurring theme as this work discusses the different methods and interpretations of probability.

Naive Definition of Probability Following Laplace, the probability of an event is defined to be the number of favorable cases divided by the number of equally possible cases. This method of defining probability significantly broadened the scope of the measure to be used for random events for which only the relative frequencies may be known. Probability defined in this manner will be referred to as Laplacean probability in this work.

Probability is a measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more likely it is that the event will occur. A simple example: in a vast majority of cases of flipping the coin, the coin will land on heads or tails and its probability of landing on either one is 1/2.

# 2. Basic Concepts

Impossibility, certainty and an impossible event – and event A is impossible if A = Ø, and event A is certain if A = S. This is easier to understand by considering a certain event, using the earlier example a certain event D on tossing a coin that it will land either on the floor or the ground. An impossible event cannot actually occur.

Occurrence of an event – An event A is said to occur if the outcome of an experiment results in a member of the set A. Using the coin example, if the result of the experiment was the coin landing on a person’s shoulder, we would say “event A = {H} has not occurred”. Note that an event can have more than happening, e.g. if we toss the coin twice, the event C = {H} can occur twice, three times or not at all.

Complimentary events – Events A and B are said to be complimentary if A = {not B} (A occurs when B does not occur). In the above case, if we define an event A = {H} and B = {T}, then A and B are complimentary, and if A occurs then we know B cannot have occurred and vice versa.

Event – Any subset of outcomes from an experiment is called an event. Using the example of tossing a coin, an event A = {H} is getting a head, and event B = {H, T} is the coin landing on the table (an outcome which is impossible in real life, but we’ll assume it can happen…).

Experiment, outcome and sample space – An experiment is the process that produces the outcome. The set of all possible outcomes of an experiment is called the sample space (often denoted by S). For example, the experiment of tossing a coin has the possible outcomes ‘Heads’ and ‘Tails’, which can be listed in the sample space S = {H, T}. The outcomes in a sample space are usually elementary and mutually exclusive events.

# 3. Probability Distributions

The probability distribution of a random variable is a description of how the probabilities are distributed over the values of the random variable. When we know the probability distribution, we know exactly how likely each value of the random variable is. There are two types of random variables, discrete and continuous. They differ primarily in the types of variables they use. A discrete random variable is one which may take on only a countable number of distinct values and thus can be quantified. A continuous random variable is one which takes an infinite number of possible different values. It is not defined at specific values. The main difference is that to specify a known random variable, we need only specify its probability distribution for a discrete variable. This is in contrast to a continuous random variable for which we need to specify a probability distribution and an equation to determine the probability of an event. In general we use a plot to describe the probability distribution of a random variable. Usually a histogram is the best plot for a discrete variable such as a Poisson or Binomial but for a continuous random variable the continuous probability distribution is best described by a curve or line.

# 4. Conditional Probability

This might make sense because in a relative sense, 0.6 is three times greater than 0.2, so it seems that probability has worked out. However, because we can have a contradiction between P(A|B) and P(A^c|B) (the probability that it came from the other bag given that it’s sour), probability can sometimes lead to illogical conclusions. This is why it’s so important to make sure that you’ve set up independent events or conditional probabilities might give you a false sense of confidence in your answer.

P(A|B) = P(B|A)*P(A) / [P(B|A)*P(A) + P(B|A^c)*P(A^c)] = (0.6)(0.5) /[(0.6)(0.5) + (0.4)(0.5)] = 0.3/0.5 = 0.6.

Let A be the event that the candies come from the original bag (the one with P(sour) = 0.6) and let B be the event that I get a sour candy. Then, P(A) = 0.5, P(B|A) = 0.6 and P(B|A^c) = 0.6. The probability that I get a candy from the original bag given that it’s sour is P(A|B). Using some basic algebra, we can find:

Let’s do a little thought experiment. Suppose I have three sour candies and two sweet candies, and I tell you that I got these five candies from two different bags of candy, and I’d also tell you that the number of sour candies that I get is independent of the bag from which I got them. If you also know that a bag of candy that I have contains five candies: three sour and two sweet, then you can analyze the probability of given events quite easily.

# 5. Applications and Examples

It is worth learning to recognize problems that are probability exercises. They often fall into certain general types, as in the examples below, which include card games, genetic inheritance, flipping coins, or making tests with binary outcomes. The types of problems may be different, but the probability model will always be similar, generally based on the sample space and an assignment of probability to certain events.

Whether you are interested in games of chance or discovering the likelihood of various real-life events occurring when certain conditions are specified, the subject of probability will provide you with a mathematical tool that will help you make these decisions. In this section, we will consider some simple examples and problems that give rise to probability questions. We will show how the mathematical theory applies in each case and consider whether the theory is a realistic model or whether there are other factors to take into account. Finally, we will consider some real-life situations in which the ability to calculate probability can be an asset.