Binomial Probability Distribution and the Battle of Gettysburg

Joshua Chamberlain (1828-1914). See question #11 in the Gettysburg Trivia Quiz.
Joshua Chamberlain (1828-1914). See question #11 in the Gettysburg Trivia Quiz.

This post shows how I use my Battle of Gettysburg Trivia Quiz to help teach the binomial probability distribution. My brother Roger and I are Civil War buffs, and we recently paid another of our many visits to the Gettysburg Military Park Battlefield in Gettysburg, Pennsylvania. On each visit to a Civil War battlefield, I’m rewarded with another gold nugget of information about a particular battle that relatively few people know. After several visits to the Gettysburg battlefield, I created a Gettysburg Trivia Quiz to demonstrate the binomial probability distribution in my statistics classes at the junior college where I have taught a variety of math courses over the years. I consider the binomial probability distribution to be the most important discrete probability distribution in statistics. (The normal probability distribution is the most important probability distribution in all of statistics.)

You can download the Gettysburg Trivia Quiz by clicking the links below. I believe you’ll find it fun, interesting and helpful to take the quiz before continuing.

Gettysburg Trivia Quiz (student version)

Gettysburg Trivia Quiz (teacher version)

You can also download the quiz on our free instructional content page under the statistics tab.

The features of the quiz and how I administer the quiz are as follows:

  • The quiz has 20 multiple choice questions.
  • Each question has 5 choices.
  • Each question is designed so that no normal person would have any idea of what the correct answer is. Hence a normal test taker can only guess at the correct answer, and therefore has a 0.2 probability of answering any question correctly.
  • All questions are independent of each other in that knowledge of any one question can’t help answer any other quiz question.
  • On the day of the lesson, I announce that the class will be taking 20-question multiple choice trivia quiz on the battle of Gettysburg.
  • I distribute a copy of the quiz to each student, and tell them there is no reason to take a peek at a neighbor’s answer because their quiz score has no influence on their course grade.
  • I tell them that the intent of the quiz is to eventually help them better understand the binomial probability distribution. Most likely there will be no high scores.
  • I allow 5-10 minutes to take the quiz. (You will observe intense expressions on some faces.)
  • I verbally provide the correct answers so that each student can grade his/her own quiz. I ask students to be honest when grading their quiz. There should be no surprise if almost all students have a very low score.
  • As each student reports his/her quiz score, I record the score on the chalkboard.
  • I then begin a discussion of the binomial probability distribution, introduced previously, and how the binomial distribution can be used to predict the probability distribution of scores from the Gettysburg Trivia Quiz. The class will see that the expected average or mean score is 4, and an unusually high score is 8 or more. A score of 8 is more than 2 standard deviations above the expected mean score of 4, where the expected standard deviation of the scores equals 1.789.
  • Using our calculators, the probability of a score of 0 = 0.0115, probability of a score of 4 = 0.218, and the probability of a score of 8 or more = 0.0321.
  • We then compare the experimental binomial results from the quiz with the expected binomial results. Especially with a larger class, students are amazed at how the expected and experimental results agree.

My free handout, Summary of Common Probability Distributions, describes the key properties and some applications of the probability distributions found in lower level college statistics courses. You can also download the handout on our free instructional content page under the statistics tab. Teachers can copy parts or all of the handout to share with students. Here is a general description of the key components of a binomial probability distribution:

  • A Bernoulli trial is any experiment that has exactly two possible outcomes. Examples: 1) Tossing a single coin has outcomes of head or tail. 2) Answering a multiple choice test question has outcomes of correct answer or incorrect answer. 3) Consider the chances of a person living to retirement age of 65. That person can die before 65, or age 65 or after.
  • The two possible outcomes of a Bernoulli trial are called ‘success’ or ‘failure’. The terms success and failure are just labels that represent the two possible outcomes of a Bernoulli trial.
  • Each outcome of a Bernoulli trial has no influence on subsequent Bernoulli trials. In other words, Bernoulli trials are independent events.
  • The random variable x of a binomial population equals the number of successes found in n Bernoulli trials repeated under identical conditions. Therefore x could equal any of the integers that range from 0 to n.
  • The probability of success on each trial is always the same and is denoted by p. The probability of failure on each trial is denoted by q. The laws of probability tell us that p + q = 1 or q = 1 – p.
  • The expected average or mean of random variable x equals μ = np.
  • The expected standard deviation of random variable x equals σ = √(n*p*q).
  • The formula for the probability of exactly x successes in n trials is P(x) = nCx * px * q(n-x).

Now let’s see how the binomial probability distribution can be used to predict the key statistical properties of the sample of quiz scores. Note that the number of test takers has no influence of the expected values for μ and σ below. As the number test takers increases, we can expect increased agreement between expected and experimental results.

  • Each quiz question is a Bernoulli trial; the answer is correct or incorrect.
  • The trivia quiz is composed of 20 independent Bernoulli trials with p = 0.2 and q = 0.8 because we are assuming that normal students don’t know any of the details of the battle of Gettysburg.
  • The number of successive independent Bernoulli trials equals n = 20.
  • Let the random variable x equal the number of correct answers reported by test takers.
  • The expected mean or expected class average score μ = np = 20*0.2 = 4 correct answers.
  • The expected standard deviation of quiz scores σ = √(npq) = √(20*0.2*0.8) = 1.789.
  • An unusually high score = μ + 2σ = 4 + 2*1.789 = 7.58 = a score of 8 or more.
  • Calculate the sample mean x-bar and sample standard deviation s of the reported scores, and then compare these values with μ and σ.
  • P(x = 0) = 0.0115, P(x = 4) = 0.218), and P(x ≥ 8) = 0.0321. Compare these probabilities to the experimental probabilities derived from the reported quiz scores. Example: What is the experimental probability that a student got a score of 4?

Based on the Gettysburg Trivia Quiz, here are the results of two probability simulations of a binomial distribution where the number of Bernoulli trials n = 20 and the probability of success on each trial p = 0.2. The widths of the histograms bars = 1 unit. Both the heights of the blue bars and the area of the blue bars equals P(x) = the expected probability of a student having exactly x successes in 20 Bernoulli trials. The heights of the red bars and the area of the red bars equals the experimental probability of a student having exactly x successes in 20 Bernoulli trials. The first graph shows the results of 50 students taking the trivia quiz, and the second graph shows the results of 50,000 students taking the quiz. Every simulation run on any number quiz takers will give us different experimental results, but the same expected results.

Simulated Results of 50 Quiz Takers
Simulated Results of 50 Quiz Takers
Simulated Results of 50,000 Quiz Takers
Simulated Results of 50,000 Quiz Takers

Using the same simulation parameters for the two graphs above, the graph below shows the cumulative results after a run of 30 simulations in manual mode where the user is required to press the <R> key to run the next simulation. The output of each simulation in manual mode shows us how well a test taker did on the quiz. The graph below includes a string of S’s and F’s that indicate successes and failures for test taker number 30 in 20 Bernoulli trials. The probability simulation software used to create the histograms allows me to find expected and experimental probabilities of events by just moving the mouse cursor over a histogram bar, and then shift-left clicking the mouse. The software also makes it easy to find probabilities such as P(x ≤ 5) or P(x ≥ 2).


My Probability Simulations software was used to create the graphics in this post. I will be releasing a full and free version of this software in the near future at the Math Teacher’s Resource website. The software release will be announced in a future post. Stay tuned!

I will close this post by telling you a true story about Larry (not his real name) who was in a general education high school math class that I taught approximately 20 years ago. Larry had very little mathematical ability. Larry could learn how to solve a specific type of math problem when the problem was presented in a specific format. If the presentation of a math problem was modified in any way, Larry was lost.

Like myself, Larry was a Civil War buff. Many of his classmates thought Larry was a history genius because he could spout a wide variety of American history facts. Larry could tell you that Private Hugh White was the lone British sentry on guard duty near the British Customs House on Monday, March 5th, 1770, the night of the Boston Massacre. I loved to share Civil War stories with Larry. He liked to ask trivia questions to stump me. It turns out, Larry, his father and I also had the same barber, Joe. About 6 years after Larry graduated, I asked Joe to have Larry take my Gettysburg Trivia Quiz the next time he got a haircut. I gave Joe the answer key so that he could grade the quiz while Larry was still in the shop. I told Joe that if Larry scored 19/20 or 20/20 on the quiz, I would pay for his haircut. Larry scored 19/20! The haircut and tip were on me. Joe told me how excited Larry became, and how proud Larry’s father was. I guess our brains are wired differently.

Why is Division by Zero Forbidden?

Zero1What is 5/0? When I ask my beginning algebra students that question, the most popular incorrect answer they give me is 0. The next most popular incorrect answer is 5. After repeated reminders by their math teachers, students eventually learn that 5/0 is undefined, has no value, or is meaningless. (I once told a class of 9th grade algebra students that if they use their calculator to divide a number by zero, the calculator will explode in their face. One student looked at me and said, “Really?” I forgot how literal 9th graders can be. At least I got the student’s attention.) When I ask college algebra, trigonometry, statistics, technical math or calculus students why a number divided by zero is undefined, I either get an answer that begs the question or students say it’s simply a mathematical fact that they learned in a previous course.

So how do you explain division by zero? There are two ways. The first depends on a basic understanding of division of two numbers. It goes something like this: Students learn that a / b = c if and only if a = b*c. Therefore 986 / 58 = 17 because 58*17 = 986. Is 5 / 0 = 0? No, because 0 * 0 ≠ 5.   Is 5 / 0 = 5? No, because 0*5 ≠ 5. Since 0 times any number never equals 5, 5 / 0 is NOTHING or undefined. So what about 0 / 0? The problem here is that 0 times any number equals 0, and therefore 0 / 0 would have infinitely many answers, which in turn would be rather confusing. So we say that any number divided by zero is undefined.

The second explanation involves a deep mathematical insight from the 12th century Indian mathematician and astronomer, Bhāskara II, who developed the basic concepts of differential calculus. The 17th century European mathematicians, Newton and Leibniz, independently rediscovered differential calculus. This second explanation due to Bhāskara II goes something like this. Consider a single piece of fruit. If we divide 1 piece of fruit by ¼, we get 4 pieces of fruit. If we divide 1 piece of fruit by 1/10,000, we get 10,000 pieces of fruit. As 1 is divided by smaller and smaller numbers that approach zero, the number of pieces of fruit increases without bound. Therefore 1/0 = ∞ and, in general, n/0 = ±∞ if n does not equal 0.

Bhāskara II, Newton and Leibniz discovered the revolutionary concept of a limit of a function at a point, which enabled them to get around the problem of division by zero. Once that problem was solved, it was a relatively easy task to find methods to calculate a rate of change over a time interval of length zero, rate of change over a fleeting instant of time, or rate of change over a flux of time, as Newton would say. In The Ascent of Man, Dr. Bronowski tells the viewer, “In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.” Differential calculus is all about the mathematics of variable rates of change. I should mention that differential calculus students learn a slick technique for finding the limiting value of an x-variable expression as x approaches a constant k and the value of the expression when x = k is 0/0 or ∞/∞.

The graphic below shows the graphs of the functions y = 2Sin(x) and y = 2Csc(x) along with its vertical asymptotes. The graphs are color coded green, blue and red respectively. Because Csc(x) = 1 / Sin(x), the Csc(x) function is undefined at precisely those values of x where Sin(x) = 0. It’s interesting and fun to advance a trace mark cursor on the graphs of these functions. On both graphs, the horizontal velocity of the trace mark is constant, but the vertical velocity of the trace mark changes as the value of the x changes. As x approaches a vertical asymptote, the trace mark races towards ± ∞. Differential calculus gives us a complete understanding of the phenomena of the moving trace cursor.


The above graphic, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. The equations entered into the program were: y = 2Sin(x), y = 2Csc(x), and Sin(x) = 0. Go to to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.

Differential calculus is not only interesting and fun, but it can also be a stress reliever. At least it was for Omar Bradley, the famous American WWII general. He took a calculus book with him on battle campaigns, and when opportunity allowed, he worked differential calculus problems to relieve the stress of a battle campaign.

Lincoln, Gettysburg, and Geometry

381px-Abraham_Lincoln_O-77_by_Gardner,_1863In my last blog, I showed a connection between Lincoln’s Cooper Union speech and Euclid’s Elements. There is also a connection between the fundamental postulates in a mathematical geometry and the fundamental postulates in a system of government or political geometry.

All mathematical geometries have a core set of postulates from which all statements or theorems about the relationships between the objects in the geometry are derived. Different sets of postulates result in different sets of geometric theorems. For example: in the geometry of a flat surface, Euclidean geometry, there is a parallel line postulate from which we can deduce that the sum of three angles in any flat surface triangle equals 180 degrees. In the geometry of a sphere, all lines intersect and therefore there is no parallel line postulate. As a result, the sum of the three angles in any spherical triangle is greater than 180 degrees and less than 540 degrees.

All political geometries, likewise, have a core set of postulates from which the laws of the political geometry are derived. The opening lines of the second paragraph of the Declaration of Independence stated the fundamental postulate of the new American democracy, “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.” This new American political geometry was radically different from previous political geometries.

The meaning of the word “all” in the fundamental postulate of American democracy eventually led to the Civil War. Initially, Lincoln viewed the Civil War as a struggle to preserve the Union. Only later did he see it as a struggle to make the fundamental postulate of American democracy a reality for all Americans. Students of mathematics come to fully appreciate and understand terms like “all,” “each,” and “every” only after they have acquired a sufficient level of mathematical maturity. Lincoln had acquired sufficient political maturity by the time he gave his Gettysburg Address on November 19, 1863.

Lincoln began his Gettysburg Address by reminding his audience of the fundamental postulate of American democracy. The Civil War was about making this postulate a reality for all Americans. He then tells Americans that we, the living, have the responsibility to finish the work of the living and dead Americans who so nobly fought to advance the fundamental postulate of American democracy. The struggle goes on today.

All human enterprises are connected. Whether the enterprise involves art, music, literature, dance, theater, mathematics, physics, chemistry, biology, political science, computer science, history, psychology, religion, sports, etc., there is always some connection to be found. When I find a connection that I have not realized before, I experience one of life’s special moments, a moment of epiphany.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have you experienced?

Visit us at

Photo credit: “Abraham Lincoln O-77 by Gardner, 1863” by Alexander Gardner – Library of Congress. Taken on November 8, 1863, just 11 days before Lincoln’s address at Gettysburg.