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# Lesson Plans for Teaching Binomial Distributions

The first few years teaching AP Statistics, I enjoyed teaching students about binomial distributions, but I didn't quite understand how it was connected to the bigger picture. It always felt like an "add on" that we cover so that students can answer a few questions that will likely show up on the AP Exam. With more experience, I realized that binomial distributions are a critical step in the progression towards the end goal of the course.

## A Path to Inference

Here is the realization that helped me bridge the gap from binomial distributions to inference:

A binomial random variable measures the count of the number of successes in a fixed number of trials. Any inference about categorical variables uses the proportion of successes in a fixed number of trials. In order to change the count of successes into a proportion, simply divide by the sample size. Put simply, a sampling distribution of sample proportions is a scaled version (by 1/n) of a binomial distribution.

This means that any of the properties of binomial distributions are also properties of the sampling distribution of sample proportions. Also the formulas for binomial distributions can be easily scaled to become the formulas for the sampling distribution of sample proportions. Here are some ideas that we get from binomial distributions that we will use later with sample proportions:

• 10% condition

• Large Counts condition

• Formulas for mean and standard deviation for the sampling distribution of sample proportions.

By the end of teaching this content, we can even ask students real inference questions:

Mr. Wilcox gets a bag of candy that contains 200 Skittles and he gets only 13 red Skittles. Does this provide convincing evidence that the company's claim that 15% are red is a lie?

## Four Lessons to Teach Binomial Distributions

First, set the stage for these lessons by teaching Days 53 - 56.

### DAY 1: Is it Smart to Foul at the End of the Game?

• Determine whether the conditions for a binomial setting are met.

• Calculate and interpret probabilities involving binomial distributions.

Darius Washington is fouled at the end of a basketball game and has a chance to win the game for Memphis if he can make three free throws. Students develop an understanding of the binomial formula by considering the probability that Memphis wins.

### DAY 2: Pop Quiz!

• Determine whether the conditions for a binomial setting are met.

• Calculate the mean and standard deviation of a binomial random variable. Interpret these values in context.

This lessons starts with a pop quiz for students. The catch is that there are no questions, and they simply have to guess the answer for every question. Students then use their prior knowledge of random variables to arrive at formulas for the mean and standard deviation for a binomial distribution.

### DAY 3: Where Are All the Green Skittles?

• Calculate and interpret probabilities involving binomial distributions.

• When sampling without replacement, check the 10% condition so that independence of trials is satisfied.

• When appropriate, use the Normal approximation to the binomial distribution to calculate probabilities.

Several years ago, Skittles changed the flavor of the green Skittle from lime to sour apple (big mistake!). In this lesson, students arrive at two of the conditions that we will need later for inference: the 10% condition and the Large Counts condition.

### DAY 4: How Many Bottle Flips To Go Viral?

• Find probabilities involving geometric random variables.

• Calculate the mean and standard deviation of a geometric random variable. Interpret these values in context.

So everyone knows about the bottle flip challenge. But where did it come from? Answer: It was a viral video of a high school student who successfully performed the challenge at the high school talent show. What you don't know was that he did a lot of his practicing in his AP Stats class. Be sure to show this video at the end of the lesson. Thank you to all-star AP Stats teachers Kelly Pendleton and Alana Braland who designed this lesson at Stats Medic Summer Camp. It is absolute fire. Fun fact, the student in the video that went viral ACTUALLY WAS Kelly's student when the video went viral. This created a ton of buy in from our students when we told them about our friend Kelly and her student. Follow Kelly and Alana on Instagram @srsly.statistics (best handle ever!).

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