Slides: Fallacies of Probability (Dr. Feddes)

 Sources for this presentation

• Daniel Kahneman, Thinking, Fast and Slow
• Rolf Dobelli, The Art of Thinking Clearly

 

Airport closed

What is more likely?
A. Chicago airports are closed. Flights are canceled.
B. Chicago airports are closed due to bad weather. Flights are canceled.

Vincent

Vincent studies the Bible a lot. He prays for others. He’s kind. He has leadership gifts. He is a talented speaker. What is more likely?
A. Vincent is an accountant.
B. Vincent is an accountant and a bi-vocational pastor.

Conjunction fallacy

Judging the conjunction of two things as more likely than one of the things

• A and B is never more likely than A and never more likely than B.
• A conjunction can never be more probable than either of its conjuncts.

Why is this fallacy attractive?

• Representativeness: extra details that match a mental stereotype feel more plausible, BUT each added item decreases probability
• Plausibility blinds us to probability

Ask the experts

What is more likely:
A. Oil consumption will decrease by 30 percent next year.
B. A dramatic rise in oil prices will lead to a 30 percent reduction in oil consumption next year.

Daniel Kahneman asked this at an international conference of academic experts. B was considered much more likely. Even experts commit the conjunction fallacy.

Car probability

James has a gorgeous wife, owns a big house, drinks fine wines, and wears expensive clothes. What car is James more likely to drive?
A. Ferrari
B. Toyota

Toyota. A thousand Toyotas are sold for every Ferrari.

Work probability

Susan wears glasses, has a degree from a university, reads poetry and novels, and loves classical music? What is Susan more likely to be?
A. English professor
B. Stay-at-home mom

Stay-at-home mom. There are hundreds of stay-at-home moms for each female English professor.

To estimate probability correctly for James and Susan, you must pay attention to the base rate.

Base rate fallacy

Focusing on details of specific case but ignoring general base rate

When you hear hoofbeats behind you, don’t expect a zebra!

Base rate errors can come from:

 

Medical base rates

• Abdominal pain can mean cancer,  ulcer, flu bug, or indigestion.
• Doctors, even specialists, are trained to check for most common problems before rare conditions.
• Sometimes likelihood of diagnosis is much lower than testing error.

Pastoral base rates

Some causes of bizarre behavior:

• Biochemical imbalances
• Drug abuse
• Relational wounds
• Demon possession

Check for causes with higher base rate before looking for demons.

Base rate fallacy

Base rate errors can come from:

• Intuition jumping to conclusions
• Inability to figure some math

To calculate probability of an event, we must include base rate data and do math correctly.

Virus testing

• A virus has infected 5% of people.
• Amy feels fine but gets tested for this virus in order to be safe.
• Amy’s test comes back positive.
• A test is correct in 90% of cases.

For every 1,000 people, 45 true positives and 95 false positives

   true positives for virus    .
(true pos.) + (false positives)

          (.05 x .90)                   .045    
(.05 x .90) + (.95 x .10)   .045 + .095

Taxicab problem

• A city has two cab companies, Green and Blue. 85% are Green; 15% Blue.
• A cab was in a hit-and-run at night.
• A witness says the cab was Blue.
• Witnesses are 80% correct discerning between Blue and Green cabs.

For every 100 claims to see Blue, 12 will be true; 17 will be mistaken.

     true positives for Blue   
(true pos.) + (false positives)

           (.15 x .80)                  .12     
(.15 x .80) + (.85 x .20)     .12 + .17

Kidney cancer

A study of kidney cancer in the 3,141 counties of the United States found that the lowest rates of kidney cancer were in counties which are mostly rural, sparsely populated, and in traditionally Republican states. Why?

A study of kidney cancer in the 3,141 counties of the United States found that the highest rates of kidney cancer were in counties which are mostly rural, sparsely populated, and in traditionally Republican states. Why?

Small numbers fallacy

• Rural lifestyle can’t be the cause of extreme lows AND extreme highs.
• What’s the real “cause”? Greater variation in smaller sample sizes.
• Extreme outcomes (both low and high) are more likely to be found in small samples than in large samples.

Billion dollar blunder

• The most successful schools tend to be among the smaller schools.
• Gates Foundation spent $1.7 billion to learn why small schools were best.
• The least successful schools tend to be among the smaller schools.
• “Cause” was small numbers fallacy

Small numbers fallacy

• We seek causes, certainty, and stories that make sense.
• We pay more attention to the content of messages than to information about their reliability.
• Causal explanations of statistics go wrong if “cause” is small sample. 

Scold or praise?

“If I praise super performance, the person usually does worse the next time. If I scold poor performance, the person usually does better. Don’t tell me reward works better than punishment. Clearly, scolding gets better results than praise.”

Regression to the mean

• Below-average performance improves closer to average the next time.
• Above-average performance drops closer to average the next time.
• Such regression to mean is normal, and does not prove anything about effectiveness of praise or scolding.

Very smart women

A. Very smart women tend to marry men less smart than they are.

B. Correlation between intelligence scores of spouses is not perfect.

Regression to the mean

• Depressed children taking pills (or hugging cats, or drinking tea) improve over a three-month period.
• Extremes regress to the mean over time, regardless of other factors.
• In testing, the treated group must improve more than control group.

Regression to the mean

• “He was fantastic in Game 1 but not in Game 2. He must have crumbled under the pressure.”
• “I felt so close to God last month, but now I don’t feel as spiritual. I must be doing something wrong.”
• Extremes tend to return to normal!

Gambler’s fallacy

Assuming that unrelated events influence each other.

• “My coin flip came up heads twice in a row. It’s due to come up tails.”
• “We’ve had three girls in a row. If we have another baby, it’s almost sure to be a boy.”

Neglect of probability

• Lottery players focus on size of jackpot, not likelihood of winning.
• After news of plane crash, travelers cancel flights; probability remains miniscule that theirs will crash.
• Amateur investors compare funds’ yields but not levels of risk.

Induction overconfidence

Thinking and acting as though past events are safe guide to future

• Goose eats day after day: “Farmer has my best interests at heart.”
• Thrill seeker: “I’ve done 1,000 crazy jumps. I’ve never been hurt.”

Induction and probability are helpful, but only as long as current trends continue.

• Black Swan: improbable, unforeseen event that changes everything
• Invention, war, pandemic, disaster

Induction and probability address short-term unknowns, not final certainties.

• Today your death is unlikely. But your death is certain (unless Jesus returns first).
• This year Jesus’ return is unlikely. But Jesus’ return is certain.

Fallacies of probability

• Conjunction fallacy
• Base rate fallacy
• Small numbers fallacy
• Regression to mean fallacy
• Gambler’s fallacy
• Neglect of probability
• Induction overconfidence