Introductory Statistics and Analytics: A Resampling Perspective

4

PROBABILITY

“Is this a game of chance?”

“Not the way I play it.”

(W. C. Fields)

Until now, we have been dealing regularly with the notion of probability without really defining it; in this chapter, we introduce more formal concepts of probability. After completing this chapter, you should be able to:

• produce a Venn diagram,
• use the addition rule and explain in what circumstances it is relevant,
• calculate weighted means and expected values,
• calculate standardized values (z-scores),
• interpret the role of the standard Normal distribution as a benchmark.

What is Probability?

Most people have an intuitive sense of probability and an idea of chance. The weather forecaster does not need to explain what is meant by “chance of rain.” Formal and more scientific understandings are more elusive—volumes have been written on probability over the centuries, in the realms of history, philosophy, and mathematics. So, we will not get tangled up in formal definitions of probability. Instead, we will point to two useful concepts in interpreting probability:

1. Long-run frequency: Probability can be seen as the frequency with which an outcome would occur of the event, which could be repeated over and over again. For example, the proportion of times you will get “heads” if you flip a coin over and over again. This is easiest to understand for a concrete process, such as a game of chance, whose repetition ...