[I had some formatting problems putting this post together, which was causing some text to disappear. Fixed now.]
Patterico poses a statistical question:
50 standard coins are thrown onto the floor. Before the toss, each had an equal chance of coming up heads or tails. Before you see how they came up, all 50 coins are covered up.
Tom goes and uncovers 25 of them. All are heads.
Joe offers to give you $10 for every remaining coin that came up tails, if you will give him $12 for every remaining coin that came up heads.
1) Do you accept Joe’s offer?
2) Is there any question you would like to ask Tom before you decide?
The “official” answer has not yet been posted.
My answers, slightly edited, behind the spoilers tag:
Show ▼
My question to Tom is, “Did you selectively uncover only the coins showing heads?”
An affirmative answer reveals a very great deal of new information. The selection of remaining covered coins is no longer random, and assuming the coins and the tosses are honest, as claimed, I’d take the bet.
If, on the other hand, he randomly selected the coins to uncover, and they all came up heads, I’d take that as evidence that something was rotten — either the coins or the tosses or something was not random, and I’d expect all or most of the remaining coins to be heads as well, and I wouldn’t take the bet.
We can restate the problem thusly:
A bit generator, alleged to be random, is tested 25 times; each time it issues a one.
What will the next test show?
The chances that the bits are truly random, that each bit is truly independent of the previous bits, is very small.
It’s possible that you’ve just watched a 1:33 million (that is, 1:225) “miracle” — but the way to bet is that the bits are not random, not independent, and that the 26th bit will also be a one.
In the current case, the source of the bit stream in question is not just the coin-flip; Tom is essentially acting as the display for the generator. We are testing the entire system, generator and display, not just the generator.
While the coin flip is said to be honest, Tom has the chance to filter the output, and no allegation has been made concerning Tom’s honesty. (Indeed, we are explicitly invited to question his honesty.) With 25 heads in a row, you deserve whatever penalty Joe assesses if you assume that Tom has not imposed some bias on the (originally) random bit stream.
The whole point of statistical testing is to check for systemic biases. Suppose you are doing quality control at a widget factory. Widgets are built from doohickies and gizmos, which have independently been tested before assembly. Out of each lot of a thousand widgets, you select 10 samples (I’m making these numbers up.) Normally, you might find one or two bad widgets in a given sample. Today, however, all ten widgets in the sample fail the instant power is applied.
Do you:
A) Assume you’ve just witnessed the one time in the entire history of the plant where the sample consists of all the bad widgets in the lot, and initial the QC report “OK”? (Raise your hand, everybody who said “Don’t take Joe’s bet, and don’t bother to ask Tom any questions; he’s probably a nice guy.”)
Or
B) Fail the entire lot, and notify the shift supervisor that something has gone awry on the assembly line at or after the point where doohickies are connected to gizmos?
A surprisingly large number of very intelligent and well-informed people went for option A. They assumed the bit stream, from coin flip to display, was completely random, and under that assumption, they’re correct: even after a string of 25 heads, the next flip has a 50/50 chance of turning heads. This is essentially a statistically-savvy knee-jerk, responding to what’s known as the “Gambler’s Fallacy”: “I’ve had a bad run, but if I stay in the game a little longer, my luck is bound to turn.”
They failed to notice Tom.
This entry was posted
on Saturday, August 23rd, 2008 at 11:53 am and is filed under Throwing Out the Trash.
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