Statistics for Data Science and Analytics

5Probability

“Is this a game of chance?”

“Not the way I play it.” W. C. Fields

Source: Courtesy of Universal Studios Licensing LLC.

We have been dealing regularly with the notion of probability; 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
• explain the binomial probability distribution
• calculate standardized values (z scores)
• interpret the role of the Normal distribution as a benchmark

5.1 What Is Probability

Most people consider that they 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 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 of the event would occur if repeated over and over. For example, the proportion of time you would get “heads” if you flip a coin many times. This is easiest to understand for a concrete process, such as a game of chance, whose repetition ...