Random Variables, Expectations, and More About Games

Abstract

Suppose we toss a coin three times in independent trials, with probability p of getting a head at each trial. On the average, how many heads can we expect to get? Right now this question does not have a precise meaning for us; what do we mean by “average” or “expect”? Most likely we do have a rough idea of the meaning of the question. Three tosses of a coin will result in anywhere from 0 to 3 heads, so the answer must be some number in that interval. The first step in making the question precise is the definition of the term random variable. A random variable is a correspondence that assigns to each outcome in a sample space a unique number. (Mathematicians more generally refer to such objects as functions.) For example, in the above setup of three tosses of a coin, let X = the total number of heads obtained The table below shows the assignment of a value for X to each of the eight possible outcomes.