Teaching the Binomial Distribution with Sports

In this edition of Teaching Math with Sports, we look at the binomial distribution. The binomial distribution is one of the simplest yet most ubiquitous models that are used throughout statistics. Understanding when and how it can be applied is key to teaching the binomial distribution.

The binomial distribution is used to describe “yes or no” outcomes (hence binomial). In sports – when things are easily categorized into success or failure – this is a natural distribution to use in many situations This is why the binomial distribution in sports is so common.

In this article we include three examples showing how the binomial distribution can be applied to solve sports analytics questions. The point is not to necessarily be an all-inclusive lesson for teaching the binomial distribution, but to provide examples of applications of this probability distribution.

Binomial Distribution Background

The binomial distribution is applicable when a two-outcome event is repeated N times and the probability of each outcome happening is the same across each event. The canonical example is flipping a coin N times and counting the number of heads. This is a binomial distribution because we repeat the exact same event (flipping a coin) N times and the probability of heads is the same on each flip.

Typically one of the two outcomes is called “a success” while the other is called “a failure”, even if neither outcome is necessarily considered good or bad. In the previous paragraph, we would call flipping heads a success and flipping tails a failure.

A binomial distribution is characterized by two parameters: N , the number of times the event is repeated, and p the probability of success on each trial. A binomial distribution with these parameters is denoted B(N,p) .

Example 1: On-Base Percentage in Baseball

Every time a batter comes to the plate in baseball, they either (a) get on base or (b) do not. Plate appearances in baseball are an example of a “two-outcome” event which can be repeated. Therefore, the binomial distribution is the exact correct probability distribution to use when considering questions about “how many times a baseball player gets on base in a set number of plate appearances”. Let’s see an example.

Joey Votto’s career on base percentage is 0.412. Suppose somebody asks you how many times you expect Joey Votto to get on base in his next 50 plate appearances. The number of times Votto gets on base in the next 50 plate appearances follows a B(50,0.412) distribution.

Challenge Question 1: Why is it common for player’s to average .400 over a short stretch of games but nearly never over the course of an entire season?

Example 2: Why Longer Playoff Series Favor the Better Team

The NCAA and NCAAW college basketball tournaments are widely loved because of their unpredictable nature. Almost all of the madness and the frequency of upsets comes from the fact that it is a single elimination tournament. That means that only one game is played between the two teams and the winner of that one game advances to the next round. In the NBA or MLB, playoff winners are decided by series. As a result, the favorite advances to the next round much more often. But why?

The winner of a playoff series can be modeled using the binomial distribution. A series length has to be an odd number, call it 2N+1 . Let the favorite have probability p of winning an individual game. Then, the probability that the favorite loses the series can be computed by using the cumulative distribution function for the binomial distribution B(2N+1,p) . In particular, if \phi is the CDF for B(2N+1,p) , then the probability that the favorite loses is \phi(N) .

Let’s see how the probability of the favorite winning a playoff series changes with the length of the series. On the x-axis below is the length of a hypothetical playoff series. On the y-axis is the probability that the favorite wins the entire series. The different color dots represent different levels of how heavily favored one team is.

Notice how as the series length increases, the probability of the favorite winning the entire series also increases. This is intuitive. It is more difficult for an underdog to win 4 times in 7 games than it is for an underdog to have one good night.

Challenge Question 2: Why shouldn’t you use the binomial distribution to simulate series where home field advantage has a significant effect on the probabilities of teams winning?

Challenge Question 3: In the NBA finals, a team wins the championship if they win 4 games out of 7. However, if a team wins 4 out of their first 6, the 7th game is not played. This means that the number of experiments/trials in our binomial distribution might not be a fixed number. Does this mean that we shouldn’t model series in sports with the binomial distribution?

Example 3: Free Throws and Fouling at the Ends of Games

At the end of basketball games, fouling by the losing team is a common tactic to try to snatch a victory from the jaws of defeat. If a team shoots 80% from the free throw line, then the points scored by the shooting team can be modeled by a B(2,0.8) distribution.

At the end of games, announcers are often heard saying things like “This team doesn’t need a 3 here, an easy two would be better!”. We want to use the binomial distribution to determine whether or not this is the case.

Here is the setting for our hypothetical: Team A is beating Team B by 1 point. Each team will get the ball two more times. Therefore, Team A takes 4 free throws and the number of points they score can be modeled by B(4,0.8) .

Team B has the choice between an easy two pointer (75%) or a three pointer (40%). If they take the two point shot both times, their points scored can be modeled by two times a B(2,0.75) distribution. If they shoot threes, their points scored is three times a B(2,0.4) distribution.

Which type of shot gives them the best chance at winning?

Challenge Question 4: Considering conditional probability distributions, what happens if Team A misses both their free throws on their first trip to the line? What if they make one? Both?

Example 4: Being Careful About Assumptions

One key caveat in using the binomial distribution is to make sure that the number of trials N and the probability of success p cannot change during the course of the event. If these things do change, then the binomial distribution is not the correct tool to use.

Here is a “non-example” of when to use the binomial distribution in sports; students can learn from this example by being asked whether or not the binomial distribution is an appropriate model.

We consider football and kicking extra points. Suppose that on a given weekend, 50 extra points are kicked and the league average extra point accuracy is 95%. Can the number of field goals made be modeled by a B(50,0.95) distribution?

The answer is No.

Of those 50 extra points, some are kicked by good kickers while others are kicked by below average kickers. Thus, on some of the trials/experiments the success rate might be 98% while on others the success rate might be closer to 90%. Therefore, the probability of success changes from trial to trial.

One of the best tricks anyone can learn for improving their mathematical thinking is “taking things to the extreme”. That is, if a situation seems like a toss-up and you can’t tell, try changing the numbers and making them more extreme to see what happens. Here’s how you can do this to help understand this example better.

What if there were two kickers, one was 100% accurate and the other was 0% accurate. If they each kick 2 field goals, then their average field goal accuracy will be 50%. However, we should not model the number of field goals that are made with a B(4,0.5) distribution. In our example, the good kicker will make both of this field goals while the bad kicker will make neither of his. Therefore the probability of 2 kicks being made is 100%. If you modeled this scenario using a B(4,0.50) distribution, the probability of 2 kicks being made would only be 37.5%.

Challenge Question Answers

Challenge Question 1: Compare two alternatives. Suppose that a particular player gets a hit 30% of the time they come to the plate (they hit .300). Which is more likely: them hitting .400 over the course of 50 at bats or over 500 at bats?

To hit .400 over 50 at bats, you need to record 20 hits. This can be calculated using the CDF of the B(50,0.3) distribution. To hit .400 over 500 at bats, you need 200 hits. Compute this probability using the CDF of the B(500,0.3) distribution and compare the results.

Also check out this related article about some effects of a shortened baseball season.

Challenge Question 2: Because the probability of a team winning at home versus on the road is different. Therefore, the binomial distribution is not an appropriate model. The best approach here is to model the number of wins for each team by a sum of two different binomial distributions (one for each game location) with different p values.

Challenge Question 3: The series stops early because the outcome is already decided. The winner won’t change if we go to the full 7 game series. Said another way, the binomial distribution doesn’t care about order.

Challenge Question 4: Instead of using a B(4,0.8) distribution, use a B(2,0.8) distribution and add either 0, 1, or 2 to the results depending on the outcome of the first set of 2 free throws.

To subscribe for email updates when new articles (especially related to teaching!) are posted, use the form below.