Home 

Time-Axis Fallacy and Bayes Theorem

Proposal
Number Sense
Interactive Quiz
Lesson Plans
History
Problem Bank
Glossary
Quotes
Helpful Links
References


Introduction: Most students understand that the probability of an event occurring can be influenced by another event that has already occurred.  However, many students cannot understand that the probability of an event occurring can actually be dependent on an event that occurred later.  Having information about the outcome of an event can be used to revise probabilities of the occurrence of a previous event.  This lesson plan will help teachers to correct this common student misconception.  For students with more probability experience, this lesson also introduces them to Bayesí Theorem.  Bayesí Theorem provides a formula that to find one conditional probability if other conditional probabilities are known.  More specifically, it can be used to find P(A|B) if P(B|A) is known.  Bayesí Theorem is usually expressed as:

 

P(AB)  [P(B|A) * P(A)]
P(A|B) = ----------- = --------------------------------------------
P(B) [P(B|A) * P(A) + P(B|~A) * P(~A)]

I. Probability Topic(s): Time Axis Misconception, Dependence of Events, Tree Diagrams, and Bayes' Theorem

A. NCTM Standards addressed and link to NCTM: Algebra, Data Analysis & Probability, Problem Solving, Reasoning & Proof, Communication, Connections, and Representations.  http://standards.nctm.org/protoFINAL/chapter7/introduction.html

B. Related Connections: Medicine  

 

II. Assumed prior experience with: Complementary probabilities, Dependent events, the formula: P(A|B) = P(AB) / P(B).

III. Rationale: To correct a common misconception about probability (known as the Time-Axis Fallacy) and to introduce Bayes' Theorem. 

IV. Learning Objectives:

1. Students will learn that the knowledge of an eventís outcome can affect the probability of the unknown outcome of an event that has already occurred.

2. Students will learn how to apply Bayes' Theorem to solve probability problems.

V. Materials & Technology Needed:

1. Bag and four jellybeans for modeling (optional).

2. Calculators for mathematical computations.

VI. Procedure:

PART 1: INTRODUCTION

1. Present the following situations to your students and ask them the questions.  For all situations, suppose that there are four jellybeans left in a bag.  Two are grape and two are licorice 

2. Situation 1:  If I reach in to the bag without looking and choose a jellybean at random, what is the probability that it is a grape one?  What is the probability that I choose a licorice jellybean?

3. Situation 1 (continued):  Suppose that I choose a grape jellybean.  I reach into the bag again to choose another jellybean at random.  What is the probability that I pull out the other grape one?  What is the probability that I take a licorice one?  Why do the probabilities change from what they were before?

PART 2: TREE DIAGRAM

4. Represent this situation as a tree diagram.  (For students who are unfamiliar with tree diagrams, some explanation and terminology might be necessary.  For the rest of this lesson, I will refer to the nodes and the branches of tree diagrams.  For every event that has more than one possible outcome, the tree diagram will have a node.  Any possible outcome at a node is represented by a branch.)

5. What does the first node of the diagram represent?  How many branches does it have?  How do we label them?  What probabilities do we assign to each branch?

6. Follow the branch that assumes that the first jellybean I take is grape.  What does the node at the end of this branch represent?  How many branches does it have?  How do we label them?  What probabilities do we assign to each branch?

7. Repeat these steps for the other branch of the first node (the one that assumes that the first jellybean I take is licorice).

8. The tree diagram now has four possible final outcomes.  Label them.  How can you find the probability associated with each of these branches?

9. Suppose I start back at the beginning, with two grape jellybeans and two licorice jellybeans in the bag.  Situation 2: This time I choose one jellybean at random and put it in my pocket without looking at it.  I choose another jellybean at random and it is grape-flavored.  What is the probability that the jellybean in my pocket is also grape?  What is the probability that it is a licorice jellybean?

10. Why isnít it equally likely that the jellybean in my pocket is either flavor when there were two of each flavor in the bag when I took it?

11. Letís use the formula P(AB) = P(A|B) * P(B) to find these probabilities (the ones from Procedure 9).  What probabilities do we want to know?  What else do we need to use the formula?  Which of these do we already know?  How can we find P(G2) from the tree diagram?  Now, use the formula.

12. Letís use the tree diagram to help us find these probabilities.  Are there any branches that are not necessary in this situation (Situation 2)?  Why?  What branches are still possible?  Is there a special relationship between these two events?  What is the relationship between the probabilities of these events?  What probabilities can we assign to two complementary events where one event is twice as likely to occur as the other event?

PART 3: BAYES' THEOREM

13. Thereís a mathematical formula that can help us answer this kind of question called Bayes' Theorem.  This formula is most useful when we need to find P(A|B) but we can only easily find P(B|A) and P(A).

Formula from Bayes' Theorem:

P(AB)  [P(B|A) * P(A)]
P(A|B) = ----------- = --------------------------------------------
P(B) [P(B|A) * P(A) + P(B|~A) * P(~A)]

14. Referring back to the second jellybean situation, what probabilities are we looking for?  What probabilities do we know from the Tree Diagram?  Use Bayes' Theorem to find the two probabilities weíre looking for.  Are the answers the same as we found from the Tree Diagram?

15. A common application of Bayes' Theorem is medical testing for diseases.  Doctors perform tests to help determine if their patients have certain diseases.  The tests indicate that the person either has the disease (a positive result) or does not have it (a negative result).  Based on studies, doctors know the probability of a random person from a population having a particular disease.  They also know the probabilities of a person with the disease getting a positive test result and of a person without the disease getting a negative test result. 

16. In the fictional town of Atlantis, doctors have developed a test for the Atlantis Death Flu.  They know that the probability that a random person in Atlantis has the Atlantis Death Flu is .01.  They also know a person who has the Atlantis Death Flu has a .95 probability of testing positive for the disease while a person who does not have it has a .94 probability of getting a negative test result.  Letís say that a patient comes to a doctor because he wants to get tested for Atlantis Death Flu.  What is the probability that he actually has Atlantis Death Flu is he tests positively for it?

17. Letís represent this situation with a tree diagram.  What should appear first in the tree diagram: whether or not a person has the disease or whether or not a person tests positive for the disease?  Why?  How do we label the two branches of the first node?  What probabilities go with each branch?  How do we know the probability that somebody in Atlantis does not have the Atlantis Death Flu?

18. Follow the branch that assumes that a person has the disease.  How do we label the two branches of this node?  What probabilities go with each branch?  How do we know the probability that somebody with the Atlantis Death Flu tests negative?

19. Repeat these steps with the node at the end of the branch that assumes that a person does not have the Atlantis Death Flu.

20. What probability are we looking for?  What probabilities do we know from the Tree Diagram?  Use Bayes' Theorem to answer the question.

VII. Assessment:

1. Assume I have a normal deck of cards.  I draw a random card from it and place it face down on the table without looking at it.  Then I draw a second card and I see that it is a red card.

(a). Explain how I now have more information about the color of the first card because I know what color the second card is.

(b). What is the probability that the first card is red?  What is the probability that the first card is black?

(c). Draw a Tree Diagram to represent this situation.  Label your diagram, and if you use any abbreviations, explain what they stand for.

(d). Use Bayes' Theorem to check the probabilities you derived in part (a).

2. Use Bayes' Theorem and the probabilities from the Atlantis Death Flu situation to find P(~A|+) and P(A|-).

Time-Axis Fallacy and Bayes Theorem Solution Guide