9.1 Heuristics

Learning Goals

[Link to PlayPosit]

[Link to instructional video]

More details: [link to topic on SSS website]

After this lesson, students should

1. [What they should understand, feel, learn, etc.]

Definitions

• Term

  Explanation.

Examples

• [item 1]

Common Misconceptions

• Mistaken quote

  Explanation.

Context

[Explanation of how the current topic into the larger context of the course by explaining how the previous topic(s) relate to the current topic and leaving cliffhangers for future topics where relevant throughout the lesson]

Before

X.X Lesson Name

Explanation.

• Itemized explanation.

After

X.X Lesson Name

Explanation.

• Itemized explanation.

Recommended Outline

Before Class

• [Any essential logistical things that need to be done for this class]
• Prepare a seating chart.
• Review PlayPosit and discussion questions and ask faculty, Gabriel, or Emlen any questions you have.
• (Optional) Prepare a presentation.

During Class

• (5 min) Come up with some fun way to assign the roles of spokesperson and notetaker (e.g. earliest birthday in the year, lives furthest from campus). Remind them of the responsibilities of these roles.
• ([#] min) [Description of some module from #Lesson Content]
• (5 min) Collect questions for plenary.

After Class

• [Any essential logistical things that need to be done as followup for this class]
• Collect answers from notetakers for the forum / plenary.

Lesson Content

Clicker Question

You're sitting in your room cramming for an exam when your roommate decides to throw an impromptu party and invites people from all over the university. Realizing that you're not going to get any work done, you decide to make the most of it and start mingling with the crowd. In the process, you strike up a conversation with someone named Jessie who starts droning on and on about rockets. They go on for so long that you start to lose interest and begin thinking about what sort of person Jessie really is.

Here's some helpful numbers:

• There is one engineering major for every three non-engineering majors at your college.
• Half of all engineering majors like rockets.
• A sixth of all non-engineering majors like rockets.

Which is more likely?

1. Jessie is an engineering major.
2. Jessie is a non-engineering major.
3. It's equally likely that Jessie's an engineering or non-engineering major.

KOALA-21

We introduce and practise Bayes' rule (formulated in terms of odds) in this activity.

Clicker Question

1. Suppose there is an epidemic of KOALA-21 breaking out among koalas in New South Wales, which about 1% of the koalas have contracted. A test for KOALA-21 was developed, whose false positive rate is 9% and false negative rate is 10%. If a particular koala has tested positive of KOALA-21, what is the actual probability that it really has KOALA-21?
   1. Between 0-25%
   2. Between 25-50%
   3. Between 50-75%
   4. Between 75-100%

Instructions

1. Reminder of what true/false positive/negative rates mean. Emphasise that [math]\displaystyle{ \text{(true positive rate)} = 1 - \text{(false negative rate)} }[/math], and [math]\displaystyle{ \text{(true negative rate)} = 1 - \text{(false positive rate)} }[/math].
2. Show and explain the following diagram.
3. Introduce the idea of odds, as opposed to probabilities. Examples: 50% probability of a coin flip translates to 1:1 odds; 90% probability translates to 9:1 odds. Bayes' rule is simplest when phrased in this manner.
4. The main equation we will use is

   [math]\displaystyle{ \text{(Odds that one actually has the disease)} = \text{(Prior odds of having the disease)} \times \frac{\text{(True positive rate)}}{\text{(False positive rate)}}. }[/math]

5. In the case of KOALA-21,

   [math]\displaystyle{ \text{(Odds that the KOALA has the disease)} = (1/99) \times \frac{1-0.09}{0.10} \approx 1/9.9. }[/math]

6. We just wrote this as a ratio. But, as odds, we represent it as 1:9.9.
7. The actual probability of having the disease is [math]\displaystyle{ 1/(1+9.9) \approx 0.092 }[/math].

Discussion Questions

Have students work in small groups to work out this problem, following the KOALA-21 example above. Give assistance where needed.

1. Based on a daily case count of 73,000 (November 2021) and a 14-day recovery period, it can be Fermi estimated that the prevalence of Covid-19 is 0.3%. For the commonly used PCR test, the true positive rate (sensitivity) is 98%, and the true negative rate (specificity) is 80%. If you do one PCR test and get a positive result, what are the actual odds that you have Covid?
2. From the odds you obtained from the first test, if you do another PCR test and get a positive result, what are now the actual odds that you have Covid?
3. Why is it often recommended for first-time positive patients to test for Covid a second time?

Jessie Again

Now that we have some practice working out Bayesian odds with diseases, it's time to try and figure out who Jessie really is. Recall our "disease" formula but somewhat generalized.

[math]\displaystyle{ \text{(Odds that a positive test is accurate)} = \text{(Base odds of positivity)} \times \frac{\text{(True positive rate of the test)}}{\text{(False positive rate of the test)}} }[/math]

We're using the term "test" here very broadly. For example, our original conversation with Jessie counts! In that case, we were using Jessie's interest in rocketry as a "test" for whether or not they're an engineering major. If Jessie is interested in rockets and is also an engineering major then we have a true positive. But if Jessie isn't an engineering major then it's a false positive.

Here are the values from the original problem. Given this information, what are the odds that Jessie is an engineering major?

1. There is one engineering major for every three non-engineering majors at your college.
2. Half of all engineering majors like rockets.
3. A sixth of all non-engineering majors like rockets.

Conjunction Fallacy

Andy is a junior at Berkeley. His favorite book is Howard Zinn’s “People’s History of the United States.” He’s passionate about politics, and he regularly attends local protests. He is most likely to be:

1. A computer science major.
2. A computer science and also a political science major (double major).
3. A computer science major who is a member of the Berkeley College Democrats.
4. A computer science and also a political science major (double major) who is a member of the Berkeley College Democrats.

Representativeness Heuristic

Mini-lecture about the following examples:

1. Linda bank teller.
2. Kings day parade attack.
3. California natural disaster.
4. Michigan/Detroit.

Collect Questions for Plenary

(5 min) Collect remaining questions from the students for faculty in plenary (can be questions for clarification, extension, discussion, etc.), and add [ here].