Common Core Standards: Math
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
"Give me liberty, or give me death!"
Learning about the independence of events isn't always as exciting as learning about the independence of the American colonists. You'll certainly grab the attention of the students if you slam your fist on your desk at the beginning of this lesson, shouting the words of Patrick Henry.
With a sample space, we can take advantage of probabilities for the sample space. There are two ideas here that students should learn. The first is the idea that two events are not necessarily independent. For example, the probability of you slamming your fist on the desk could be related to the probability of your coffee spilling all over the students graded homework. For clumsier teachers, these probabilities may not be related.
The first idea is one that draws upon assumptions about relationships. Students should be able to recognize that two events are not necessarily related, even if they appear to be. At least, they should be able to recognize this under the umbrella of statistics.
Just because you give a group of third graders vanilla ice cream cones doesn't mean they are happy. Maybe they're happy because it's Friday or because Arbor Day is a week away. (Honestly, who doesn't love Arbor Day?) You want to relate to the students that there are many possible effects that on an outcome, and statistics are a useful tool to determine which events are independent.
How do we discern the clumsy teachers from the over-animated ones? We need a statistics litmus test—say that five times fast—to help us out. If we have statistics exploring spilled coffee and fist slamming in math teachers, we can use the probabilities to test the independence of the two events. This is the second idea the students should learn.
On a given day, the probability of math teachers in Virginia spilling their coffee, event C, is P(C) = 0.25, the probability of slamming their fists, event F, is P(F) = 0.10, and the probability of both occurring, event B, is P(B) = 0.025. Since P(C) × P(F) = 0.025 = P(B), we can conclude that the C and F are independent events. The Virginian math teachers surveyed aren't so zealous with their fist-slamming as to spill their coffee. It turns out that some of the teachers must just be klutzes.
When a similar survey was completed in Ohio, probabilities P(C) = 0.32, P(F) = 0.09, and P(B) = 0.08 were found. P(C) × P(F) = 0.0288, not P(B). We know events C and F are not independent in the group of sampled Ohioan math teachers. The spirit of Patrick Henry appears to be roaring in their bellies.
Feel free to have fun with this lesson. Make an outrageous claim that all carnivorous dinosaurs prefer to wear purple, polka-dotted jumpsuits. Use the probabilities of each to prove or disprove your outrageous claim. This will open the minds of the students to the idea that things aren't always what they seem. Then teach them this probabilistic tool to decide for themselves.