Introduction and Overview

This e-book is for students in STA258: Statistics with Applied Probability at the University of Toronto Mississauga. It gives you a practical sense of the ideas from lectures and tutorials. It shows how the mathematics of probability helps us talk clearly about uncertainty in everyday life and in scientific work.

Uncertainty touches almost everything we care about. Will tomorrow’s lecture move online because of a snowstorm? Can a spacecraft launch if crosswinds rise past a safe limit? Will a new medication shorten recovery time outside a clinical trial? These questions look different on the surface, yet the same statistical tools help with each one. If we treat an unknown outcome as a random variable, we can describe its typical behaviour, work out the chance of unusual events, and update those chances when new information arrives.

We all do rough versions of this in our heads. Dark clouds make us grab an umbrella, and a traffic report makes us leave early. Intuition helps, but it is easy to mislead ourselves. Streaks appear at the casino even when the dice are fair, and vivid stories can drown out quiet data. Formal probability gives a firmer base. Meteorologists can combine radar images, satellite readings, and decades of storm records to say not just it might snow, but there is a 70% chance of more than 10cm by morning. Engineers can measure how often an alloy fails under stress and state the expected lifetime of a bridge girder with a clear margin of error rather than a hopeful guess. In both cases, the numbers come from a sampling distribution. Think of it as a catalog of results we would see if we could repeat the process again and again.

Throughout this book we return to that catalog. We begin with random variables and their distributions. We then learn how to simulate, visualize, and summarize those distributions with real data. From there we build the core tools for inference. We use confidence intervals to place reasonable bounds on unknown parameters. We use hypothesis tests to weigh evidence for competing explanations. We use regression models to study the influence of several factors at once. The tone is friendly and driven by examples, so you will see snowfall totals, pulse rates, and exam scores alongside the necessary notation. Every result still rests on solid mathematics. By the end, you should feel comfortable turning a messy, uncertain situation into symbols, using probability to work through the problem, and explaining the answer in plain language to someone with no statistical background.

That habit of moving between raw experience and quantitative insight sits at the heart of STA258. It is what this e-book aims to develop.