High School: Statistics and Probability
Conditional Probability and the Rules of Probability HSS-CP.A.1
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
For many students, studying statistics can seem about as exciting as watching grass grow in the middle of winter in the northernmost reaches of Siberia. Watch out for the frostbite.
But like it or not, statistics are ubiquitous to the world around us. Like most things in math, we can relate an otherwise monotonous idea back to something that excites in the inner most children in ourselves (and the students). Doing so, we can wrap an otherwise mundane lesson in a pretty present that the students can't wait to open at 5 AM on Christmas morning.
Before we can begin answering the tough questions, we need to tackle how to pick a sample from a larger group. Next, we need to break it down into smaller sample groups that we can learn something from.
If we are asked to find the average number of female aliens that land on U.S. territory each year, we don't want to sample a population of little green visitors to Argentina. We put on our logoed T-shirts and bring our clipboards, and we survey a group of American alien tourists as they step off their saucers. Among other questions, we inquire if they are male or female.
Once we have a valid sample, we want to break it down into groups that are meaningful to our question. Put another way, we want to list possible outcomes that can answer the question at hand. In this case, we'd want to break our sample into two groups: male and female. Assuming that aliens either male or female, breaking them into these groups exhausts all possible outcomes for the sample.
With a sample, we can sort the set of possible outcomes using operations "and," "or," and "not". These ideas can easily confuse even MENSA members in large enough combinations, so we need to address these ideas carefully to the students.
Sometimes, we get indirect information about a sample, which means we have to sort it in a way that answers our question. We want to group the sample into those that do and do not have a certain characteristic. We also want to connect the idea that, just because they do not belong to one group doesn't automatically admit them to another group. For example, just because a Skittle is not orange doesn't mean it's green.
The "or" operator is a way to join sets of samples that are otherwise unrelated. Students should be able to recognize that this is a way of including two or more unrelated groups of outcomes. It represents the union and has the symbol ∪. Students can remember this because the ∪ symbol represents a ∪nion.
Meanwhile, the "and" operator is the evil twin of the "or" operator. The "and" operator also joins two sets of samples, but this time they must already be related by a common characteristic. In other words, so satisfy the "and" operator, two samples must have an intersection. This is given the symbol ∩, for i∩ntersection.
It's simple to visualize all three of these operators using Venn diagrams. Given a sample with two different sets of characteristics, you can sort them with either the "and" or the "or" operator and to get very different outcomes.
The "and" and "or" operators can also be combined with the "not" operator (which is given the tilde symbol, ~). This is an advanced idea that will almost certainly require a Venn diagram. Once you get to this point, just walk the students through the steps individually. For example, if you have a bag of red, green, and blue marbles, if a marble is not red or green, it is blue.
Since there are many ways to combine these operators to sort sets, you should walk your students through simple examples one at a time, building the complication step by step using Venn diagrams. Using different colors to represent the different groups can be instructive.
It's also easy to get caught up in the ambiguity of the individual operators as words. "Mrs. Applebottom, did you say 'or' or 'and'?" Writing the statements on the board may help to clarify some of the confusion as the students learn the concepts.
Finally, keep in mind that this information can be boring by nature. Choose an example sample (rhyme time) that your entire class can identify with, or their heads will be hitting their desks before you finish the third sentence. Consider introducing an entertaining sample before the topics to whet their appetites and grab their attention.
- Combinations and Probabilities Brain Teaser - Math Shack
- Venn Triagram Analysis - Math Shack
- Venn Diagram Analysis - Math Shack
- Unions of Sets - Math Shack
- Word Problems Involving Sets - Math Shack
- Complementary and Mutually Exclusive Events - Math Shack
- Intersection of Sets - Math Shack
- Venn Diagrams - Math Shack