10Combined and Conditional Probabilities

Introduction

All of the probabilities calculated up to this point have been, in a sense, simple probabilities. This doesn't mean that they were necessarily easy to calculate or understand, but rather that they all share some fundamental properties which will be explained below. In this chapter, we will talk about some not‐so‐simple probabilities, most of which won't be that hard to calculate and understand. We need to introduce some notation that can capture the ideas that we'll be presenting in a concise and precise manner.

Functional Notation (Again)

In Chapter 2, we introduced the idea of a function and then went on to Probability Distribution Functions. The common notation for a function is to use a letter followed by a variable, or variables, in parenthesis. Since we're dealing with probabilities, we'll stick with convention and use the letter P. If we're talking about rolling a pair of dice, we would represent the probability of a random variable, x, with the notation P(x).

Then, the probability of the roll yielding 12 is

(10.1) This notation is a little bit confusing in that if we wanted to multiply a variable P by 12, we could write it the same way. The result of course most likely wouldn't be 1/36, depending on the value of the variable, P. How do you know whether we're talking about a function or a multiplication? From context ...

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