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.
INTERPRETATION OF OBJECTIVE - S.CP.B.8
This objective has builds on the previous more basic multiplication rule. We begin to not only interpret this relationship, we begin to make calculations.
(1) The student will be able to calculate probabilities using the General Rule of Multiplication, P(A and B) = P(A)P(B|A) = P(B)P(A|B).
(2) The student will be able to understand the General Rule of Multiplication through Venn diagrams.
THE BIG IDEA
We are now ready to expand from the basic multiplication rule, P(A) • P(B) = P(A and B) to the more complex one that includes all possible relationships, independent or not. P(A and B) = P(A)P(B|A) = P(B)P(A|B) handles both independent and dependent relationships.
TRAPS & PITFALLS
There is just simply a lot to keep track of..... most of it comes down to the student's understanding of independence. If they have a sound grasp on the two tests for independence then they should be fine. Also the key words of replacement or no replacement help student greatly in knowing what to do.
Classically these types of problems are done in a deck of cards but I find that fewer and fewer students know what is in a deck. They tend to get problems wrong, not because of the math, but because of their lack of experience with the cards in a deck. I have moved to more problems with marbles. They seem to be easier to understand.
This relationship is really what we have built all the previous concepts for. Here we use conditional probability and independence to make the correct decisions about how to use the formula.
This establishes the General Rule for Multiplication of events. It is used in a variety of different locations.
MY REFLECTIONS (over line l)
At this stage all of the previous objectives lead us to general rule. It allows for the calculation of independent and dependent events. Focusing on independence helps students know which of the two formulas to use and keeping the wording consistent is very helpful such as replacement or no replacement. I like to do some actual examples of pulling from an actual bag of marbles to demonstrate these differences.