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The French bistro omelette

From https://books.google.co.jp/books?id=5tjdvIQeFB0C

There was once a little family bistro outside Paris. It acquired a reputation for the great lightness, flavour, and delicacy of its omelettes. A group of connoisseurs were writing up a compendium of French cuisine. They decided it was essential to include a recipe for these famous omelettes.

The mother of the house kindly obliged. She gave them a detailed recipe of ingredients and the order and manner in which they were blended as the omelette was prepared. There was only one snag. When the Parisian chefs tried it, they got a different result. They found delicious flavours but their omelettes were totally lacking the lightness that the little family bistro achieved.

Frustrated, they experimented. They interpreted the recipe in various ways. They tried different forms of omelette pan. Eventually, they decided that there must be a trick. The mother had surely not revealed all her secrets!

So, off they go to the bistro. There, they find that the mother has died. Her daughter now cooks the omelettes.

However, her omelettes are just as good as her mother's. So they show her the mother's recipe and ask whether it is correct. She reads it carefully. Yes, she says, it is a marvel of accuracy, down to the last milligram of ingredient. This is exactly what she herself does. "There's nothing missing?", they ask. "Of course not", she replies, "Mother has written it all down." In fact, she says, her mother wrote it down for her exactly the same way. She is following precisely the same recipe as they were trying to follow.

They naturally ask her whether they can watch as the omelette is prepared. Of course, she says. There is nothing to hide. So they watch carefully to try to detect the slightest difference between the written recipe and what she does. And they are amazed at what they see. At the very beginning of her preparation, she separates the whites and the yolks. She blends the flavourings into the yolks and folds the beaten whites in only at the end, just before cooking.

They upbraid here. She and her mother, they say, have both provided inaccurate information. How could anyone fail to reveal this crucial fact in such a recipe? The daughter is offended. She takes them to task for their "stupidity" and "arrogance". Indignantly, she surveys the assembled dignitaries and asks them, in all innocence, "How else do you think anyone prepares an omelette!?"

Defense attorney

From Calling Bullshit

Imagine that you are a prominent defense attorney, defending a mild-mannered biologist against the charge that he has committed the greatest art heist of the new century.

The crime was a spectacular one. A wealthy collector shipped her private collection of thirty European masterpieces, by guarded railcar, from her home in Santa Clara to an auction house in New York City. When the train reached its destination, the boxes were taken to the auction house to be unpacked. The journey had passed uneventfully and the boxes appeared untouched, but to everyone's shock, the four most valuable paintings had been cut from their frames and were missing! Neither the police nor the insurance investigators were able to find a single clue - save for one fingerprint on the frame of one of the missing paintings. The stolen paintings were never found.

Without any other leads, the police ran the fingerprint through the FBI's massive new fingerprint database until they found a match: your client. (He had provided his fingerprints to the Transportation Safety Administration in exchange for the convenience of leaving his shoes on when passing through airport security.) Upon questioning, your client was discovered to have no alibi: He claimed that he had been cut off from human contact for two weeks, tracking radio-collared grouse through the High Sierras as part of a research project.

Lack of alibi notwithstanding, you are convinced that your client could not be the culprit. He is more of a bumbling academic than a steely-nerved art thief. He already has two NSF grants to fund his work on grouse breeding and seems to have little desire for additional income. And as best as you can tell, he doesn't know a damn thing about art - he thinks Donatello is a Ninja Turtle, for goodness' sake.

Yet the case goes to trial, and you face a brilliant up-and-coming prosecutor. After presenting all of the other evidence against your client - weak and circumstantial as it may be - she moves on to her trump card, the fingerprint. She describes the computerised fingerprint matching process to the jury, and closes her statement by declaring "There is absolutely no chance of getting a fingerprint match this good simply by accident."

You counter. "You said there is absolutely no chance of a match this good. That can't be possible. Every test has at least some chance of being mistaken."

"Well, sure", she concedes, "the test could be mistaken in principle. But for all practical purposes, there is zero chance of it actually happening. The FBI's own studies have shown a one-in-ten-million chance that two different fingerprints would match as well as the ones we have here. One in ten million: That's certainly far beyond any reasonable doubt!"

This is just what you were looking for. You turn to the jury and sketch out a two-by-two table on a large pad of paper. It looks something like this:

Match No match
Guilty
Innocent

"I think we call agree," you continue, "that this crime was indeed committed by someone. And let us also assume that the true culprit is in the fingerprint database. He might not be," you say to the prosecutor, "but that would only weaken your case. So let's assume for now that he or she is included."

The prosecutor nods.

"Then the table looks something like this." You will in the top row with a large red marker.

Match No match
Guilty 1 person 0 people
Innocent

Turning to the prosecutor, you inquire, "Now how many people are in that FBI database of yours?"

She breaks in: "Objection, Your Honour! How is that possibly relevant?"

"Actually, it's the heart of the matter," you respond. "I think I'll make that very clear in the next few minutes."

"Objection overruled."

Your opponent concedes that with all US criminal prints, all civil background-check prints, and all of the prints from the TSA, there are approximately fifty million Americans represented in the database. And of course the majority of those people will have prints that do not match the fingerprint on the painting's frame.

"So now we can fill in more of the table," you respond. You add a 50,000,000 to the lower right corner.

Match No match
Guilty 1 person 0 people
Innocent 50,000,000

Now you point at the lower left corner - innocent people who nonetheless match - and you ask, "What do you think goes here?"

You look right at the prosecutor. "You said that there is a one in ten million chance that the algorithm finds a false match. That means that with fifty million people in the database, there should be about five people who match the fingerprint found at the scene of the crime. So we can complete our table as follows."

Match No match
Guilty 1 person 0 people
Innocent 5 50,000,000

"So look at this," you say, turning to the jury. "The prosecutor is trying to distract you by calling your attention to this comparison." You point along the bottom row. "There is a one-in-ten-million chance of seeing a match by accident. But that's not relevant to what we're doing here in the courtroom. We don't care what the chance is of having a match, given that my client is innocent. We already know that we've got a match."

"No, we want to know what chance there is that my client is innocent given that we've got a match." Now you point to the left-hand column. We expect there to be about five innocent matches in the database, and one guilty person. So given a match, there is about a one-sixth chance that the accused actually committed the crime.

"Now, I can't prove beyond a shadow of a doubt that my client is innocent. All I've got is his word that he was following, dammit, what kind of bird was that again? Never mind. The point is that I don't have to prove that my client is innocent. Here in America, he is innocent until presumed guilty, and the standard of proof for guilt is 'beyond a reasonable doubt.' If there are five chances in six that my client is innocent, we're clearly nowhere near that standard. You must acquit."