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80-110 Nature of Mathematical Reasoning

Spring 2002

Dirk Schlimm

Homework 7

Wednesday, February 27

Due Monday, March 4

1. Bayes' Theorem (3 points)

a) Prove Bayes' Theorem: P(H|E)=[ P(E|H)*P(H) ] / P(E)
from the axioms and definitions introduced in class.

b) Assume that H stands for "The hypothesis is true" and E stands for "There is evidence that supports the hypothesis".

1. Under this interpretation, what does P(E|H) mean?
2. What probabilities do you need to know, in order to determine the value for the probability that the hypothesis is true given that there is evidence for it?
3. Describe a situation where you might want to use Bayes' Theorem.
2. Interpretations of probability (3 points)

Summarize the main interpretations of probability discussed in the entry probability of the Cambridge Dictionary of Philosophy (Handout #10). Write a few sentences for each interpretation.

3. Review: Midterm questions (2 points)

• Formulate 4 questions that you think would be appropriate to ask on the midterm exam. (Please no yes/no questions). They can cover any material discussed in class, or presented in the handouts, or in the book up to where we've read it.
• Rate the difficulty of your questions on a scale between 0 (piece of cake) and 10 (impossible to answer).
• Write answers to the questions, such that you find the answer worth of full credit.
• Motivation: The more interesting I find your questions, the more likely they are to appear on the exam...
4. Review: Concept map (2 points)

Draw a concept map that contains all concepts that you've learned in this class so far.

Put a concept within a circle and connect it to the other concepts or names. Mark the arrows indicating the relation that holds between the concepts, e.g., "is a", or "can be" etc.

You will see an example of a concept map in class on Friday.

The more concepts and connections, the better!