The Taxicab Problem

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 85% of the cabs in the city are Green and 15% are Blue.

A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?

Psychologists Amos Tversky and Daniel Kahneman offered this problem to study subjects in 1972. The right answer is about 41 percent:

  • There’s a 12% chance (15% times 80%) of the witness correctly identifying a blue cab.
  • There’s a 17% chance (85% times 20%) of the witness incorrectly identifying a green cab as blue.
  • Thus there’s a 29% chance (12% plus 17%) that the witness will identify the cab as blue.
  • And that means there’s approximately a 41% chance (12% divided by 29%) that the cab identified as blue is really blue:

https://en.wikipedia.org/wiki/File:TaxicabProblem.png

Most subjects estimated the probability at more than 50 percent, some more than 80 percent.

Tversky and Kahneman call this the representativeness heuristic: When we rely on representativeness to make a judgment, we tend to judge wrongly because the fact that a thing is more representative doesn’t make it more likely.

(Amos Tversky and Daniel Kahneman, “Evidential Impact of Base Rates,” No. TR-4, Stanford University Department of Psychology, 1981.)

| Science & Math

Literary Limericks

https://commons.wikimedia.org/wiki/File:%C3%9Altima_escena_de_Hamlet,_por_Jos%C3%A9_Moreno_Carbonero.jpg

Did Ophelia ask Hamlet to bed?
Was Gertrude incestuously wed?
Is there anything certain?
By the fall of the curtain
Almost everyone’s certainly dead.

— A. Cinna

Once a raven on Pluto’s dark shore
Brought the singular news: “Nevermore.”
‘Twas of useless avail
To ask further detail,
His reply was the same as before.

— Anthony Euwer

There once was a fellow called Hyde,
Whose twin self he couldn’t abide;
But Jekyll, the Devil,
Dragged Hyde to his level,
“Inside job,” cried Hyde, as he died.

— E.J. Jackson

When Ireland was bloody and leaderless,
The tedious, garrulous Daedalus —
Having failed both as priest
And as Glorious Beast —
Sailed away to write books that were readerless.

— Gina Berkeley

| Literature · Poems

Extempore

https://www.flickr.com/photos/tom-margie/1535708412
Image: Flickr

We never did stop ad libbing. No two performances were ever quite the same. One matinee, during the second month in New York, I cooked up a little surprise for Groucho. During one of his quieter scenes, while I was offstage, I selected a blond cutie from the chorus, and asked her if she’d like a bigger part in the show. She was willing and eager. I told her all she had to do was run screaming across the stage. She did, and I tore after her in full pursuit, leaping and bounding and honking my horn. It broke up Groucho’s scene, but when the laughs subsided, Groucho was ready to top it. ‘First time I ever saw a taxi hail a passenger,’ he said.

— Harpo Marx, Harpo Speaks!, 1961

| Entertainment

The Perpetual Diamond

This is bewildering: This diamond isn’t moving, and its luminance and texture are unchanging. Yet when it’s surrounded with very thin edge strips whose luminance changes with respect to the background, the whole diamond seems to move. Using the controls at the bottom, you can even direct the illusion to send the diamond drifting “up,” “down,” “left,” or “right.” But it ain’t moving.

See the paper below for details.

(Oliver J. Flynn and Arthur G. Shapiro, “The Perpetual Diamond: Contrast Reversals Along Thin Edges Create the Appearance of Motion in Objects,” i-Perception 9:6 [2018], 2041669518815708.)

| Oddities · Science & Math

Black and White

wainwright chess problem

A densely imaginative little problem by Joseph C.J. Wainwright. White to mate in two moves.

Click for Answer
| Puzzles

Light and Shadow

yamashita building blocks

Building Blocks, by Kumi Yamashita. “I sculpt using both light and shadow. I construct single or multiple objects and place them in relation to a single light source. The complete artwork is therefore comprised of both the material (the solid objects) and the immaterial (the light or shadow).”

More.

| Art

A Catalog of Clever Expedients

https://books.google.com/books?id=vOhIAAAAMAAJ

Henry T. Brown’s 1868 mechanical encyclopedia Five Hundred and Seven Mechanical Movements is being animated on this fascinating site — see working demonstrations of hundreds of mechanical linkages from the age of steam.

Note that it’s a work in progress — the movements that have been animated are indicated with colored thumbnails. Owner Matt Keveney plans eventually to animate all 507 movements in Brown’s original text.

(Thanks, Sharon.)

| Technology

Unquote

“Consistency is the last refuge of the unimaginative.” — Oscar Wilde

“Consistency requires you to be as ignorant today as you were a year ago.” — Bernard Berenson

“Convictions are variable; to be always consistent is to be sometimes dishonest.” — Ambrose Bierce

| Quotations