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For those who missed it, here is a link to Episode One

That diary ended with a discussion of availability as one of the key judgment biases too which we poor limited humans fall prey.

This diary will discuss the second judgmental bias/heuristic, that of representativeness.

Join me for today's lesson below the fold...

But first, a question...

Here is the set up....

Imagine that you are on Let's make a deal and are offered the choice between three doors.  Behind one door is a great prize, and behind the other two are junk.  After you have chosen your door, Monty looks at you and says...  Let's show you what's behind one of the doors you didn't choose....

Surprise, surprise, it's junk.  

Then he says..  "Would you like to switch to the other door or keep yours?"

So - which is it?


The probability that event X belongs to set Y (or X generated by Y) is judged on the basis of how similar X is to the stereotype of Y.

So, for example, when I ask my class for a stereotypical bird, they say "Robin", or "Crow", when actually the most common bird in the world is a chicken.


Well, chickens don't look like a "representative" of the bird family becuase:

  1.  They are large
  1.  They don't fly
  1.  We eat them

This judgmental error comes into play in day to day activities in a number of ways.  But first, a quick review of probability theory...

Let p(X) represent the objectve likelihood of X being true.

  1.  For any event A, 0≤ p(A)≤1.
  1.  If the set of events of a given type is denoted by S, then p(S) = 1.
  1.  Addition:  p(A or B) = p(A) + p(B) - p(A and B)
  1.  If A and B are mutually exclusive, then p(A or B) = p(A) + p(B), as p(A&B) is zero
  1.  Conditional Probability p(A|B) = p(A&B)/p(B)
  1.  Multiplication Rule: p(A&B) = p(A) * p(B|A) = p(B) * p(A|B)
  1.  Bayes' Rule used for updating beliefs with new data p(A|B) = p(A&B)/p(B)

7a. Using the multiplication rule can be rewritten as:
p(A|B) = p(A) p(B|A) / p(B)

7b. Using disjoint decomposition of p(B), this can be rewritten as
p(A|B) = p(B|A) * p(A) / (p(B|A) * p(A) + p(B|~A) * p(~A))

So, here's an example (and the answer may surprise you):

Suppose you were starting work and had to pass a drug test.  The test is very reliable:  The test will state the presence of an illegal drug 95% of the time when it is present and the test will be negative 95% of the time when no illegal drugs are present

Assume 5% of the population are regular drug users

If your test comes up positive, what is the ACTUAL probability that you are using drugs?

Let pos = event that test is positive (~pos = negative)
Let user = even that you are a drug user

We want p(user|pos)

p(user) = 0.05
p(pos|user) = 0.95 (specificity TP / TP + FN)
p(~pos|~user) = 0.95 (sensitivity TN / TN + FP)

Bayes’ calculation

p(user|pos) =

p(pos|user)*p(user) / p(pos|user) * p(user) + p(pos|~user) * p(~user)

= (.95 * .05) / (.95 * .05) + (.05 * .95) = ONE HALF.  

Just as reliable as a coin toss!

I imagine you were as surprised as most of my students.  

Why were you surprised?  Because 95% accuracy just sounds so good!  The problem is, with a low base rate (5%), it takes an EXTREMELY accurate test to move far away from that base rate.

As a rule, there are three factors to consider in estimating P(A|B)

What is the prior probability of A (base rate)?
How strong is the association between B and A?
How reliable is the information?

Most people ignore 1 and 3 and just use 2.  That is the representativeness heuristic (subtitled, people like a good story).

Another example, to see if you're paying attention.

A taxi cab was involved in a hit-and-run accident at night.  Two cab companies operate in the city, the Green and the Blue.  You are given the following data:

85% of the cabs in the city are Green, 15% Blue.

A witness identified the cab as blue.  The court tested the witness’s ability to identify cabs under the same conditions that existed the night of the accident.  When presented with a sample of cabs, (half of which were Blue, the rest Green) the witness made correct identifications in 80% of the cases and erred in 20% of the cases.

Question:  What is the probability that the cab involved in the accident was Blue?

Don't do the math right away, but what do you "think" it is.

Well - here are some results from my past classes:

Modal Response: 80%

Percentage of class responding with:

 The witness rate (80%) 43%
 The base rate (15%)     7%
 Median response      72.5%

For those of you who didn't jump right into the math, the answer is 41.3%

Why are so many people so bad at this?  They take the witness' testimony at face value and use it as their judgement.  Basically, they are confusing the probability that the witness said it was blue given that it was blue, with what they want, which is the probability of it being blue given that the witness said it was blue.

One more example, then on to something else.  This one is far more severe, and may generate some debate.

100 doctors were told this hypothetical scenario...

Suppose you have examined a woman for breast cancer.  The woman has a lump in her breast, but based on many years of experience, you estimate the odds of a malignancy as 1 in 100 (for a woman of this age with these symptoms).  Just to be safe, however, you order a mammogram.  The mammogram  correctly diagnoses about 80% of malignant tumors and about 90% of benign tumors.  The test comes back and, much to your surprise, it is positive.  

What is the probability that your patient has cancer?

Frighteningly, 95 of the doctors said: 80%

What's the real answer?  So that you don't have to do the math, it's .075 (7.5%).  So, a number of doctors would scare the crap out of this woman and 93% of the time, they would be WRONG.

Let's look at this on a frequency scale...

Of 1000 women given additional screening based upon an abnormal examination (e.g. lump)

Assumptions:  Cancer Base Rate= 8%, True positive hit rate =92% true negative 88%

CancerNo cancerTotal
Pos. test74110184
Neg. test6810816

So, what does this mean?

False Positive Rate: 110/184 = 60%.  So, you scare the crap out of a lot of people that didn't need to be scared.  Now, I am not advocating against scans, but you really have to be concerned about the accuracy of tests for small base rate events.

OK - Last bit....

But first, more laws of probability:

A compound event p(A & B) cannot be more likely than either of its components p(A) or p(B).

So, let's hear about Linda...

Linda is 31 years old single outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice and also participated in anti-nuclear demonstrations.

Please rank the following statements by their probability, using 1 for the most probable and 8 for the least probable.

a. Linda is a teacher in a primary school.
b. Linda works in a bookstore and takes Yoga classes.
c. Linda is an active feminist
d. Linda is a psychiatric social worker
e. Linda is a member of Women Against Rape
f. Linda is a bank teller
g. Linda is an insurance salesperson.
h. Linda is a bank teller and is an active feminist

Given the rules of probability, h CANNOT be more likely than c or f, however, in the general ranking of these the order goes c, h, f.  


Because people like a good story.  Linda SOUNDS more like h than c.  The more specific you are about people, the more likely others think that the description is accurate, although it cannot be so.

Anyway - I will close with a piece of advice.

When estimating x, start with the base rate of x.  The less you know, the closer your estimate should be to the base rate.  Extreme forecasts should only be given from very reliable predictors.

I will leave it to commenters to discuss why this is so important in our soundbite ADHD culture.

Originally posted to DrWolfy on Fri Feb 23, 2007 at 09:29 AM PST.


Do you switch doors?

42%9 votes
23%5 votes
33%7 votes

| 21 votes | Vote | Results

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Comment Preferences

  •  Tip Jar? (14+ / 0-)

    Well - I have been told I have to.

    -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

    by DrWolfy on Fri Feb 23, 2007 at 09:53:39 AM PST

  •  I predict there is a 90% probability that (3+ / 0-)
    Recommended by:
    walkshills, paul2port, pioneer111

    only 30% of the readers of this diary will get past the first 50% of the diary, and only 10% of those will get to the end. Sooo, what was the prize behind the door? /snark

    -6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

    by skrymir on Fri Feb 23, 2007 at 10:09:07 AM PST

    •  Yeah - probably should have done this in (1+ / 0-)
      Recommended by:

      two parts...  There is a lot of material.

      -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

      by DrWolfy on Fri Feb 23, 2007 at 10:15:41 AM PST

      [ Parent ]

      •  Might be better to use betting metaphors. (0+ / 0-)

        I found that explaining rudimentary probability  math gets much easier when the subject is betting for money. Sports betting, crap tables odds/betting, blackjack, etc. People relate to those metaphors and the subject of money always commands rapt attention.

        -6.38/-3.79::'A man is incapable of comprehending any argument that interferes with his revenues.' Descartes

        by skrymir on Fri Feb 23, 2007 at 10:58:28 AM PST

        [ Parent ]

  •  Statistics and Probability in an uncertain world (0+ / 0-)

    These are really interesting diaries and I have marked them so I can come back to them.

    High school education basically focuses on algebra, geometry, trigonometry and sometimes calculus -- all the linear mathematics.

    But there is a really valid argument that says the most important thing a high school student could learn is statistics and probability and their relevance to decision making.  I agree with that line of argument.
    In that sense, statistics and probability begins to help people deal with uncertainties in knowledge and outcomes and may even inform their philosophies (and at least weaken if not undermine theocratic certainty).

  •  You ALWAYS switch the doors (3+ / 0-)
    Recommended by:
    mvr, paul2port, DrWolfy

    The Odds of you picking the correct door initially is 1/3. That does not change after another door has been revealed to be a poor choice. Since probabilities ALWAYS add up to 1, and there are two doors unrevealed and your door has a 1/3 probability, the other door now has a 2/3 probability of being the big prize. 2/3 of the time you are better off switching. So your odds are exactly twice as good switching as staying the same. Sometimes a better example is usingh bigger numbers. Imagine playing Deal or No Deal. You pick a case, and play through until there are 2 cases left. One is for a million dollars, the other for any other ammount. The odds of your case being a million are 1/26. Revealing all the other cases did nothing to change these odds. It is why most of the time the contestants are better off taking a Deal than playing to their case,

    •  Do you have this posted outside your office (0+ / 0-)

      on the 4th floor of a new university?

      If so I missed seeing you at the mix and mingle yesterday d_avid_.

      •  nope (1+ / 0-)
        Recommended by:

        Remember it from UCSD, and use it when i teach math. Plus, it was in a Ask Marilyn column in Parade magazine. She always has good stuff on logic.

        •  It is in her book... (1+ / 0-)
          Recommended by:

          And I love the part where she prints emails from actual Math teachers who call her an idiot for getting it "wrong", when she has it right.

          -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

          by DrWolfy on Fri Feb 23, 2007 at 11:37:12 AM PST

          [ Parent ]

        •  Sorry, a case of mistaken identity (0+ / 0-)

          My friend tells this joke about himself and his math professor colleagues:

          How can you tell an extroverted mathematician?

          He's the one looking at the top of the other person's shoes.

          Bada boom.

          Try the veal I'll be here all week.

          PS Friends of Linda in Dr. Wolfy's example above get a bonus for giving the base rate of female mathematics professors. (So just how close to zero is the probability of an extroverted female math professor?)

          What the heck, I apologize in advance for telling my friend's sexist joke.

          Geez academic humor is not funny at all.

  •  I would begin with the human conclusions (1+ / 0-)
    Recommended by:

    thus highlighting the obvious mistaken judgments and the unsupported conclusions.

    One key point for me is: as humans, we seldom ever have complete information in making a decision.

    Sure, if Jane is hanging off a cliff, we give her a hand. Or if it's Scooter, we stomp on is fingers after peeing on them. (Not a perfect world illustration.) The point is that we draw on many factors that aren't easily abstracted.

    Most real human decisions involve a system of statistical probability analysis that is at least intuitive and not formally systemic. That is say, intuition, emotions and personal experience provide as much in terms of data as abstracted reasoning. Most people can't visualize multi-dimensional abstractions in motion but most can come to an emotional conclusion to some degree.

    So, working backward from a dramatic decision to the behavior decisions could prove enlightening than a more academic structure of applied logical analysis. I'm basically saying one over logical - that is, the illogical - can work in definitional terms.

    I will also state that our individual universe is merely a function of our attention and our momentary focus. Thus, I suspect we intuitively set up statistical models for problem solving whether we know the formalized (and recognized) mathematical logic behind them or not. Our brains and our being have to have an operational basis for decisions affecting its immediate survival and humans have some capabilities beyond fear or flight responses. Structures of illogic can have positive conclusions much like negative numbers in multiplication and division can.

    Finally, I seldom ever reveal the really important things I have in mind. Many times snark is closer to the real truth than my direct words. But my serious judgments lie with those really important things. Even if I know the truth doesn't mean I will reveal it to you or that you can translate it or have the power to understand it as I present it.

    I think an analysis of propaganda dispersion would be informative, for I see that as an interesting juncture of logic and illogic, whereby accepting the illogical depends on a logic method, tripwires in the jungle, as it were.

    "Please don't let me be misunderstood." heh

    Illumination is cheap around here.

    by walkshills on Fri Feb 23, 2007 at 10:45:30 AM PST

  •  Good stuff here Dr Wolfy (0+ / 0-)
    Bayes theorem rocks!!

    One minor correction though in the breast cancer example - The 60% number you cite (110/184) is not actually the false positive rate.  The false positive rate is defined as the percent of patients without disease who have a positive result on the test.  This is 110/920 (12.0%).  False positive rate is also defined as 1-specificity for the test.

    The 60% is better termed the percent of patients with a positive test who do not have disease.  This can also be defined as 1-positive predictive value.  PPV is the percent of patients with a positive test who actually do have disease, which here is 74/184, or 40%.

    I'm sure you probably already know all this but I wanted to correct the record.

    Got an issue, here's a tissue - Will & Grace

    by Flinch on Fri Feb 23, 2007 at 11:01:47 AM PST

    •  Thanks for the advice... (0+ / 0-)

      I have been calling it "false positive" since the test is positive when the results are absent.

      Maybe I should just call it type-I.


      -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

      by DrWolfy on Fri Feb 23, 2007 at 12:03:58 PM PST

      [ Parent ]

  •  Forecasting (0+ / 0-)

    For those interested I recommend:Forecasting Principles

    With regard to your last example. Even professionals get caught out. The wishful thinking forecaster is seen all the time. Recently a Canadian pollster, Allan Gregg, who is also associated with the Conservative party was quoted in the Globe and Mail Tories surge on Harper's Leadership

    Were an election to be held today, 34 per cent of voters would opt for the Conservatives, up three points from last month. The Liberals are at 29 per cent, down six points from the same period, while the New Democratic Party is down one point to 14 per cent. The main beneficiary appears to be the Green Party, which has the support of 12 per cent of voters, up from eight per cent.

    "These numbers are good enough that I would start my election preparedness in earnest," said Allan Gregg, chairman of The Strategic Counsel, which produced the poll for The Globe and Mail-CTV News. "I would say, looking at this, something untoward would have to happen for him not to call it."

    Mr. Gregg said the underlying favourability ratings may give Mr. Harper a better chance for a majority government than he had at the beginning of the campaign that led to his victory on Jan. 23, 2006. The Conservatives have moved a ways to establishing themselves as a governing party, he said.

    "The potential to move his numbers are far greater this time than they were last time," he said. "I would assume if they're looking at these same numbers, they've said we're having a spring election, and something will have to happen between now and the early part of April for us not to have it."

    The poll finds that 53 per cent of voters find Mr. Harper to be the most decisive of the four main party leaders, with 20 per cent opting for the NDP's Jack Layton.

    The Tories got hundreds of mentions of this "analysis" at home and around the world. It was picked up by Reuters, AP etc.

    Gregg was less than truthful. The poll showed the Tories had moved outside the margin of error for the first time. The next poll by another company reported the Liberals and Conservatives popularity within the margin of error, a statistical tie.

    Soundbite, ADHD indeed.

    •  Yes - Doctors are notoriously bad (1+ / 0-)
      Recommended by:


      Often time computer models do much better, and never worse.

      Amazingly, the "best calibrated" profession is meteoroligist.  

      When I do overconfidence and learning from feedback, I'll go more deeply into that.

      -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

      by DrWolfy on Fri Feb 23, 2007 at 12:02:19 PM PST

      [ Parent ]

  •  More on false positives (0+ / 0-)
    It's widely assumed, as in your breast cancer example, that false positives are harmless, at worst a little added expense. But I've talked to many people who have had a false positive and been deeply, deeply stressed by it, as they contemplated the what-ifs and made a lot of deep grooves

    Fry, don't be a hero! It's not covered by our health plan!

    by elfling on Fri Feb 23, 2007 at 11:47:48 AM PST

    •  sorry, toddlerus interruptus (1+ / 0-)
      Recommended by:
      It's widely assumed, as in your breast cancer example, that false positives are harmless, at worst a little added expense. But I've talked to many people who have had a false positive and been deeply, deeply stressed by it, as they contemplated the what-ifs and made a lot of deep grooves in their brains about their potential demise or a serious illness. Anecdotally, I've seen depression and anxiety from such an event last a decade.

      That is not to mention the mental and physical trauma caused by additional tests.

      False positives have a real cost and it's important when judging the results of a screening test that you compare the two populations in total outcome, not just deaths from the disease you think you're targetting. If you reduce breast cancer but increase death from heart attack, the overall outcome of giving the test to a large population may not be better.

      Fry, don't be a hero! It's not covered by our health plan!

      by elfling on Fri Feb 23, 2007 at 11:52:46 AM PST

      [ Parent ]

      •  I teach my decision making class (1+ / 0-)
        Recommended by:

        that not all mistakes are equally bad...

        One needs to understand the full costs of false positives and false negatives in order to set decision making thresholds.  

        We talk at length in my class about Doctors and why they tend to over-medicate things.  Not because of the true cost of a false negative to the patient relative to the cost of a false positive, but rather, patients tend to sue for false negatives.  

        So - it is a huge agency problem.

        -6.5, -7.59. All good that a person does to another returns three fold in this life; harm is also returned three fold.

        by DrWolfy on Fri Feb 23, 2007 at 12:01:10 PM PST

        [ Parent ]

        •  or, rather, :-) (0+ / 0-)
          That they believe patients will sue for false negatives.

          (And it may be true that they will. But if doctors had to offer a warranty on their services, they might be more leery of overmedicating and the like.)

          Fry, don't be a hero! It's not covered by our health plan!

          by elfling on Fri Feb 23, 2007 at 12:41:07 PM PST

          [ Parent ]

  •  On chickens, (0+ / 0-)

    You might not think that a stereotypical N would have to be the most common type of N. Perhaps it should just have most of the features that typically differentiate the majority of Ns from the majority of non-Ns. And if it were true that chickens don't fly (isn't it just that most have their wings clipped) you might think that what differentiates birds from most other non-bug creatures is that they fly and most other small to medium sized creatures don't.

    So I'm not sure that asking for a stereotypical N is the same as asking for the most common sort of N.

    I'm just quibbling about the chickens, not about the rest of the points in your diary.

    •  Good point (1+ / 0-)
      Recommended by:
      Most chickens can fly at least a little if they're not tightly confined in a cage. "Flew the coop" didn't come from nowhere. :-)

      Besides, my life experience tells me that chicken is spontaneously created in shrink-wrapped packages, and that chickens don't have feathers.

      Fry, don't be a hero! It's not covered by our health plan!

      by elfling on Fri Feb 23, 2007 at 12:43:57 PM PST

      [ Parent ]

  •  Tag suggestion (0+ / 0-)
    I suggest you add "science" to your tag list for these diaries. I think it fits with those.

    Fry, don't be a hero! It's not covered by our health plan!

    by elfling on Fri Feb 23, 2007 at 01:00:25 PM PST

  •  suggestion (0+ / 0-)

    one suggestion - you should define p(A|B).
    I have not looked at probability theory in years,
    did not know what this meant.  Of course a quick
    google told me immediately, but just want to note
    this omission given that you put in a primer....

  •  Both doors now equally likely for prize. (0+ / 0-)

    In terms of probability a previous comment said
    "initially the odds were 1-3, that did not change
    when one door was eliminated".  Of course this wrong,
    you now have more info and know that the odds are
    50-50 as one of just two remaining doors is known to
    have the prize.  

    WRT deal/no-deal, the cash value of the choice
    at any time is exactly equal to the total of
    all remaining prizes divided by the number of
    unopened cases.  an offer close to that is fair,
    and should be accepted/rejected depending on
    how cheap the offer is and how much the contestant
    wants to gamble.

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