Probability models are theoretical models of the occurrence of uncertain events. At the most basic level, in probability, the properties of certain types of probabilistic models are examined. In doing so, it is assumed that all parameter values that are needed in the probabilistic model are known. Let's contrast this with statistics. Statistics is about empirical data and can be broadly defined as a set of methods used to make inferences from a known sample to a larger population that is in general unknown. In finance and economics, a particular important example is making inferences from the past the known sample to future the unknown population. In statistics. we apply probabilistic models and we use data and eventually judgment to estimate the parameters of these models. We do not assume that all parameter values in the model are known. Instead, we use the data for the variables in the model to estimate the value of the parameters and then to test hypotheses or make inferences about their estimated values. Another way of thinking about the study of probability and the study of statistics is as follows. In studying probability, we follow much the same routine as in the study of other fields of mathematics. For example, in a course in calculus, we prove theorems such as the fundamental theory of calculus that specifies the relationship between differentiation and integration, perform calculations given some function such as the first derivative of a function, and make conclusions about the characteristics of some mathematical function. In the study of probability, there are also theorems to be proven, we perform calculations based on probability models, and we reach conclusions based on some assumed probability distribution. Often in life we are confronted by our own ignorance. Whether we are pondering tonight's traffic jam, tomorrow's weather, next week's stock prices, an upcoming election, or where we left our hat, often we do not know an outcome with certainty. Instead, we are forced to guess, to estimate, to hedge our bets. Probability is the science of uncertainty. It provides precise mathematical rules for understanding and analyzing our own ignorance. It does not tell us tomorrow's weather or next week's stock prices; rather, it gives us a framework for working with our limited knowledge and for making sensible decisions based on what we do and do not know. To say there is a 40% chance of rain tomorrow is not to know tomorrow's weather