5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Are you hungry because you thought about double-fudge cupcakes, or did you think about double-fudge cupcakes because you were hungry? It's all together possible that you have an insatiable appetite for cupcakes, and being hungry had absolutely nothing to do with it.

This modern chicken-or-egg argument is just one question that could be answered using a statistical approach. We could use a set of sample data to determine conditional probabilities from which we could determine the independence or dependence of the two events.

There are many different questions like this we could ask. The goal is two-fold. First, the students should be able to recognize these key concepts in terms of everyday language and situations. We want to engage students to think critically about relationships between two otherwise unrelated things. Just because a cause-and-effect relationship appears to exist doesn't mean there is one necessarily.

In our example, your hunger may be entirely unrelated to thinking about cupcakes. If they are related, did the hunger cause the thought or did the thought cause the hunger?

Next, the students should be able to explain these relationships. The students should first be able to parse the information into data that can be represented by two-way probabilities. The information can be represented using Venn diagrams or two-way frequency tables. Using the resources they've built, students can apply the knowledge they have of probabilities and independence of events to explore the relationships in question.

Let's think back to your seemingly inexplicable hunger. To answer your question more generally, you could survey 100 math teachers across your school district. You could ask the teachers if they prefer cupcakes or grilled cheese, as well as if they are hungry or not hungry.

From your results, you could create a two-way table. You could ask the same questions in the opposite order, as well, and create a similar two-way table.

Once you have your two-way table, you can calculate individual probabilities. From those, you can get conditional probabilities, and from those, you can estimate if two events are independent or dependent. Since you have two tables to look at, you could see if the results change noticeably if they were asked if they were hungry or not before being asked if they prefer cupcakes or grilled cheese.

Say the probability of you preferring a cupcake went down significantly if someone asked if you were hungry first. What would it say about which came first: hunger or the cupcake?

The most important key in this lesson is to teach students to think critically about the questions they want answers to. From this, students should be able to link their questions to the types of data they will gather. Finally, they should be able to assemble the data and infer relationships from the data using their knowledge about probabilities.