Discrete random variables and the binomial distribution

This is a version of the tutorial suitable for printing if you want to work out the questions at your leisure before checking the answers on the screen.

Random variables describe selected features of elements of a sample space. In this part of the course, we are only interested in quantitative random variables. From a mathematical point of view, a (quantitative) random variable is a function that assigns a numerical value to each element of a sample space. They come in two kinds: discrete and continuous random variables. A continuous random variable can take any value in an interval or in several intervals of real numbers, whereas in the case of a discrete random variable there are gaps between consecutive possible values. While some random variables are naturally discrete (for example, the number of siblings a person has), in many cases it is a matter of choice whether a given feature should be modelled as a discrete or a continuous random variable. For example, physical entities such as height, weight, temperature, pressure, etc. are "naturally" continuous random variables. But if we decide to measure, say, height of newborn humans in inches, then we will have very few possible integer values of this variable, and it is more meaningful to treat it as a discrete random variable.

Let T be the time that elapses until a given atom of a radioactive substance disintegrates. Is T a continuous or a discrete random variable?


Let T be the number of days in a given quarter when classes are in session (that is, excluding holidays and such). Is T a continuous or a discrete random variable?


For the rest of this tutorial, we will concentrate on discrete random variables. The PROBABILITY DISTRIBUTION of a random variable indicates how likely which values of the random variable are. The probability distribution of a discrete random variable can be represented by the PROBABILITY FUNCTION of this random variable. The probability function assigns to each possible value of a random variable its probability. For example, if the experiment consists of tossing a fair coin twice, and if X is the random variable that counts the number of heads that come up, then X can take on the values 0, 1, 2 and the probability function of X is given as p(0)=0.25, p(1)=0.5, p(2)=0.25.

Suppose your experiment consists of tossing a biased coin twice. On a single toss, the coin will come up heads with probability 0.6 and tails with probability 0.4. Let X be the random variable that counts the number of times heads comes up. Which of the following is the probability function of X?

Probability functions are frequently given in the form of tables. For example, the probability function of the previous question can be represeted by the following table:
x 0 1 2
p(x) 0.16 0.48 0.36


Let us consider a somewhat more involved example.

Roll a fair die twice. Let X be the number that comes up on the first roll, and let Y be the number that comes up on the second roll. Let Z = (X - Y)2. Which of the following is the probability function of Z?


An important feature of probability functions is that the values it takes are probabilities that add up to 1. This fact often allows one to determine missing values.

A random variable X takes the integer values 1 through 4. If p(X=1) = 0.1 and p(X=2)=p(X=3)=p(X=4), find p(X=4).

Random variables often allow one to express certain events succinctly. Let us look at some examples.

Let X be the number of hearts, Y the number of clubs, and Z the number of spades in a bridge hand. How can the event "X + Y + Z = 13" be expressed in words?

Let X be the number of hearts, Y the number of clubs, and Z the number of spades in a bridge hand. How can the event "X > Y + Z" be expressed in words?


Let X be the number of hearts, Y the number of clubs, and Z the number of spades in a bridge hand. How can the event "The hand contains an equal number of spades and hearts" be expressed in symbols?


Let X be the number of hearts, Y the number of clubs, and Z the number of spades in a bridge hand. How can the event "The hand contains 3 more spades than hearts" be expressed in symbols?


A random variable X takes the value 0 with probability 0.5, the value 1 with probability 0.3, and the value 2 with probability 0.2. Find the mean value of X.

A random variable X takes the value 0 with probability 0.5, the value 1 with probability 0.3, and the value 2 with probability 0.2. Find the variance of X.

Many random variables that occur in various applications have a binomial distribution. Such a distribution occurs whenever a random variable can be conceptualized as counting the number of "successes" in several independent "trials" of an experiment where the probability of success in each trial is the same. The probability of x successes in n trials with probability p of success in each individual trial is often denoted by b(x;n,p).

Questions that involve the binomial distribution can be solved by following a fixed sequence of steps.

  1. Determine what the "trials" are and what "success" in a single trial means. Remember that "success" is a purely mathematical concept and does not need to correspond to a desirable outcome.
  2. Determine the number n of trials.
  3. Determine the probability p of success in a single trial.
  4. Reformulate the question in terms of b(x;n,p)'s. This step will usually include determining what the relevant x (or x's) is (or are).
  5. Find the numerical answer using tables, the formulas given in your textbook, or a calculator. This last step is often the least problematic and will not be discussed in this tutorial.

On a multiple choice test, there are ten questions with five possible answers each. Dan knows the correct answers to five of these questions and randomly guesses the answers for the remaining questions. What is the probability that Dan gives exactly eight correct answers on this test?

Jane cought the flu on campus in Athens. She goes home for the weekend. Each of her five younger siblings has a 40% chance of catching the flu from Jane over the weekend. What is the probability that Jane will infect at most one of her younger siblings?

A fair coin is tossed eight times. What is the probability that heads comes up more often than tails?

Good luck on the quiz that is coming up this Thursday!

© Winfried Just
Last modified February 23, 1999