This chapter introduces probability. We begin with an informal definition which enables us to build intuition about the properties of probability. Then, we present a more rigorous definition, based on the mathematical framework of probability spaces. Next, we describe conditional probability, a concept that makes it possible to update probabilities when additional information is revealed. In our first encounter with statistics, we explain how to estimate probabilities and conditional probabilities from data, as illustrated by an analysis of votes in the United States Congress. Building upon the concept of conditional probability, we define independence and conditional independence, which are critical concepts in probabilistic modeling. The chapter ends with a surprising twist: In practice, probabilities are often impossible to compute analytically! Fortunately, the Monte Carlo method provides a pragmatic solution to this challenge, allowing us to approximate probabilities very accurately using computer simulations. We apply w 3 × 3 basketball tournament from the 2020 Tokyo Olympics.

In this chapter, there are two types of probabilities that can be estimated: empirical probability and theoretical probability. Empirical probability is calculated by conducting a number of trials and finding the proportion that resulted in each outcome. Theoretical probability is calculated by dividing the number of methods of obtaining an outcome by the total number of possible outcomes. Adding together the probabilities of two different events will produce the probability that either one will occur. Multiplying the probabilities of two events together will produce the probability that both will occur at the same time or in succession. As the number of trials increases, the empirical probability and theoretical probability converge.

It is possible to build a histogram of empirical or theoretical probabilities. As the number of trials increases, the empirical and theoretical probability distributions converge. If an outcome is produced by adding together (or averaging) the results of events, the probability distribution is normally distributed. Because of this, it is possible to make inferences about the population based on sample data – a process called generalization. The mean of sample means converges to the population mean, and the standard deviation of means (the standard error) converges on the value.