MED 607/I2T2 Project—5 day lesson plan on geometric probability ...

I went into GSP to make Venn GSP. It took me ten minutes to make somethingcomparable to the website, in that now I can change the area of the intersection bydragging one of the circles around, and I can change the areas of the circles by draggingthe endpoint of the radius around. I had to define intersection as arc segments, since arcsectors include other parts of the circle. So, it looks like if I can’t get to the web, I canmake a nice substitute with GSP. I can’t do the part where they click on A c and that partof S gets highlighted, but I can make that an extra credit project for somebody. I havesome sketches, Venn2.gsp & Venn3.gsp.I wanted to give the students a chance to do some free exploration on this website, so Iwould just ask some questions about how to define the conditional probability of A givenB, and what happens when the sizes associated with these events change.Day 5—Bayes theorem. Conditional probability will be part of their prior knowledgenow, it’s covered extensively elsewhere, and I want to use this as an example ofinvertibility, an extension I’ve never seen anywhere, and I think it’s very interesting.Inversion is a very powerful concept, and I’ve never seen it done in probability. We do itin algebra all the time, in calculus and geometry, why not probability?Learner Outcomes: The student will be able to define and give an example of Bayes’Theorem.Anticipatory set—The students have an entrance pass, two questions handed to them asthey enter the classroom. These are worth points and are collected in journals.1. If I know P(A), P(B), P(A & B), how do I find P(A given B)?2. Given the above, how do I find P(B given A) in terms of P(A given B)?Instruction: Direct teaching format as we explore the answers generated as part of theanticipatory set.1. P(A given B) = P(A & B)/ P(B). This is a standard definition. In words, weknow both A and B have to happen, and we’re interested in the relative size of (A & B) toB; B has become our new sample space, and we’re defining a probability as a relativearea.2. If P(A given B) = P(A & B)/ P(B), we know P(A & B) = P(A given B) *P(B). We can employ a symmetry argument to derive P(A & B) = P(B given A) * P(A).This gives us P(A given B) * P(B) = P(A & B) = P(B given A) * P(A), or P(B given A)= P(A given B) * P(B)/P(A).