Common Core Standards: Math
8. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
At 12:45 AM, you just can't take it anymore. You get out of bed and head to the kitchen to appease your empty stomach, which has been growling since 11. Spotting a loaf of bread and peanut butter in the pantry, you grab a knife and a paper towel. You slap a big glob of peanut butter on one side of the bread slice, set it on the paper towel, and drop the knife in the sink.
As you go to pick up the peanut-buttered bread, your hand slips over the paper towel, knocking the slice to the floor. What happens next is worse than your nightmare about the clowns with the freakishly long torsos: the bread lands peanut butter side down. There's no hope for the five-second rule on this one.
Your peanut-buttered bread is an example of a non-uniform probability model. If you were flipping an unbiased coin with equal outcomes for heads or tails, the probability model would be uniform. If there isn't an equal chance of it landing face up or face down, then the two outcomes do not have a uniform probability.
Students should first know the difference is between an outcome and an event. An outcome is just a possibility that occurs within the sample. An event is a group of outcomes that share a common characteristic. This difference is not obvious at first glance, but it is important that students understand it.
If we were to roll a six-sided die, we could say there are 6 possible outcomes. If we were to roll a 12-sided die, we could say there were 12 possible outcomes. If a student enters a Dungeons & Dragons tournament and needs to roll both simultaneously, an outcome is rolling a 5 and a 9 on each of the dice, respectively. One way we could group these into an event is the collection of all possible outcomes that include rolling a 5 on the six-sided die, of which there are 12.
With this difference in mind, students should be quick to recognize that there are situations in which two outcomes do not have an equal probability of occurring. It is helpful to compare two example situations, as we did with the peanut-buttered bread, to make this point.
More importantly, students should learn that the two situations, uniform and non-uniform probability models, must be handled differently. For now, students do not have the tools to address non-uniform models, but don't let that get them down. They'll learn more in an advanced class.
As for uniform probability models, they already know how to address many different issues. The most difficult concept is that of conditional probability. They know the conditional probability formula:
By multiplying through by P(A), we can derive the multiplication rule for probabilities.
P(A)P(B|A) = P(A and B)
We can also apply the conditional probability formula to P(A|B). This gives us another multiplication rule.
P(B)P(A|B) = P(A and B)
We can combine these two equations to get another multiplication rule
P(B)P(A|B) = P(A)P(B|A)
What does the multiplication rule have to do with uniform models? Well, these rules are only applicable for models uniform probabilities. Students should understand that non-uniform models require more complicated rules for multiplication.
As for the multiplication formulas, they are easy ways to calculate probabilities when we don't have the one we are looking for. Remember that the ultimate goal is to build a working knowledge about probabilities into the students' minds. You can show them that this is a convenient side-result that they can add to their probability toolboxes.