“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” — Galileo
Author: Greg Ross
Blackwell’s Bet
Two envelopes contain unequal sums of money (for simplicity, assume the two amounts are positive integers). The probability distributions are unknown. You choose an envelope at random, open it, and see that it contains x dollars. Now you must predict whether the total in the other envelope is more or less than x.
Since we know nothing about the other envelope, it would seem we have a 50 percent chance of guessing correctly. But, El Camino College mathematician Leonard Wapner writes, “Unexpectedly, there is something you can do, short of opening the other envelope, to give yourself a better than even chance of getting it right.”
Choose a random positive integer, d, by any means at all. (If d = x then choose again until this isn’t the case.) Now if d > x, guess more, and if d < x, guess less. You’ll guess correctly more than 50 percent of the time.
How is this possible? The random number is chosen independently of the envelopes. How can it point in the direction of the unknown y most of the time? “Think of it this way,” writes Wapner. “If d falls between x and y then your prediction (as indicated by d) is guaranteed to be correct. Assume this occurs with probability p. If d falls less than both x and y, then your prediction will be correct only in the event your chosen number x is the larger of the two. There is a 50 percent chance of this. Similarly, if d is greater than both numbers, your prediction will be correct only if your chosen number is the smaller of the two. This occurs with a 50 percent probability as well.”
So, on balance, your overall probability of being correct is
That’s greater than 0.5, so the odds are in favor of your making a correct prediction.
This example is based on a principle identified by Stanford statistician David Blackwell. “It’s unexpected and ironic that an unrelated random variable can be used to predict that which appears to be completely unpredictable.”
(Leonard M. Wapner, Unexpected Expectations: The Curiosities of a Mathematical Crystal Ball, 2012, following David Blackwell, “On the Translation Parameter Problem for Discrete Variables,” Annals of Mathematical Statistics 22:3 [1951], 393–399.)
Black and White
Podcast Episode 111: Japanese Fire Balloons
Toward the end of World War II, Japan launched a strange new attack on the United States: thousands of paper balloons that would sail 5,000 miles to drop bombs on the American mainland. In this week’s episode of the Futility Closet podcast, we’ll tell the curious story of the Japanese fire balloons, the world’s first intercontinental weapon.
We’ll also discuss how to tell time by cannon and puzzle over how to find a lost tortoise.
Talking
Excerpt of a letter from British general Philip Howele to his wife, Sept. 15, 1915:
It is VILE that all my time should be devoted to killing Germans whom I don’t in the least want to kill. If all Germany could be united in one man and he and I could be shut up together just to talk things out, we could settle the war, I feel, in less than one hour. The ideal war would include long and frequent armistices during which both sides could walk across the trenches and discuss their respective points of view. We are really only fighting just because we are all so ignorant and stupid. And if diplomats were really clever such a thing as war could never be. Shall I desert and see if any of them will listen on the other side? My little German officer was rather flabbergasted when the first question I asked him the other morning, when the escort had gone out and shut the door, and after I’d put him in a comfortable chair and given him a cigarette, was, ‘Now first of all do you really hate me, and if so why?’ He said he didn’t. But then later, when I asked him what we could possibly do to stop all this nonsense, he had no suggestions to make. ‘I have my ideas’ he said but somehow couldn’t express them.
Goodnight … and bless you.
P.
Application
Victor Hugo wrote The Hunchback of Notre-Dame under enormous financial pressure, leaving his table only to eat or sleep.
Finally, his daughter Adèle wrote, “On 14 January, [Notre-Dame] was finished. The bottle of ink that M. Victor Hugo had bought the first day was finished also; he had arrived in the same moment at the last line and at the last drop.
“This gave him, in that moment, the idea of changing his title and calling his novel: What There Is in a Bottle of Ink.”
Progress
What Did George the Third Know?
He never saw a match.
He never saw a bicycle.
He never saw an oil stove.
He never saw an ironclad.
He never saw a steamboat.
He never saw a gas engine.
He never saw a type-writer.
He never saw a phonograph.
He never saw a steel plough.
He never took laughing gas.
He never rode on a tram car.
He never saw a fountain pen.
He never saw a railway train.
He never knew of Evolution.
He never saw a postage stamp.
He never saw a pneumatic tube.
He never saw an electric railway.
He never saw a reaping machine.
He never saw a set of artificial teeth.
He never saw a telegraph instrument.
He never heard the roar of a Krupp gun.
He never saw a threshing machine, but used a flail.
He never saw a pretty girl work on a sewing machine.
He never saw a percussion cap, nor a repeating rifle.
His grandmother did his mending with a darning needle.
He never listened to Edison’s mocking machine or phonograph.
When he went to a hotel he walked upstairs, for they had no lifts.
He never saw a steel pen, but did all his writing with a quill.
He never held his ear to a telephone, or talked to his wife a hundred miles away.
He never saw a fire engine, but when he went to a fire, he stood in line and passed buckets.
He never knew the pleasure and profit to be derived from reading Science Siftings.
— Science Siftings, 1894
Fore!
Temporary rules adopted by London’s Richmond Golf Club during the Battle of Britain:
I had my doubts about this, but the rules are acknowledged by the club itself. While we’re at it: In A Wodehouse Handbook, N.T.P. Murphy notes two unusual tournament rules at the annual Bering Sea Ice Classic in Nome, Alaska:
- If a player hits a polar bear, he is penalized three strokes.
- If a player hits a polar bear and retrieves his ball, he is awarded five strokes.
Apparently those are local rules: In Extreme Golf, Duncan Lennard notes that officials of the World Ice Golf Championship actually wrote to the game’s governing body, the Royal and Ancient Golf Club of St Andrews, to ask what to do about polar bears, and received the ruling that “in the event of a polar bear wandering onto the ice golf course, the same safety procedure should be followed as for rattlesnakes and ants elsewhere in the world” — a free drop out of harm’s way.
“Juliet”
How did the party go in Portman Square?
I cannot tell you; Juliet was not there.
And how did Lady Gaster’s party go?
Juliet was next me and I do not know.
— Hilaire Belloc
Moving Day
Is it possible to pack six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box?
