The Epistemology of the Question Universe

Photo credit: Wikimedia Commons

When you watch the movie Hitch, you take it as a given that no one in that universe knows who Will Smith is. If you are going to question that, don’t see the movie.

You face a similar problem if you try to answer the following question:

You given a fair coin to flip one million times and get one million heads. What is the probability of your having gotten that result?

If this were a class and you’d like to get credit, you should say that there are two possible results for the first flip, two for the second, … and two for the millionth. This means there are 2^1,000,000 equally probable ways to flip a coin one million times. You only got one of those ways, so the probability is quite small: 1 / 2^1,000,000.

But what if your goal is to get the right answer rather than to get points on an exam?

Then, the problem itself needs to be scrutinized. First, note that every statement said by anyone ever is said by a fallible human and interpreted a fallible human.

So, we could (without adding or removing information) restate the problem:

A fallible human gives you what they say is a fair coin…

That phrasing doesn’t change any facts since the original problem statement didn’t say that G-d told you the coin was fair. But with this latter structure, we now realize that the fallible human probably fell hard.

We really ought not stop there. We are told that the coin was flipped one million times. Did they actually count? If flipping a coin and recording its results takes four seconds, then that’s 6.3 months of coin flipping 8 hours a day with weekends off. Would someone actually be able to do that? Or is it more likely for someone to feel like they must have flipped the coin for what felt like a million times to them?

You might object that the problem says that *you* flipped the coin. But presumably the question wasn’t written specifically for you. If you actually had flipped the coin yourself, you wouldn’t be reading a hypothetical question about it. The question is more likely directed at the full set of people who might be asked to answer it, not to you in particular. Unless you actually do remember flipping a coin for six months.

On the other hand, even if the coin was only flipped ten times, 1 / 2^10 is still a really small number: 0.000976562.

After getting the first few heads, it was probably worth questioning whether the coin is truly fair.

After ten heads, even a weighted coin becomes an unlikely possibility. A coin is unlikely to give ten heads in a row even if it is severely weighted. It’s now worth considering possibilities like the Super Bowl XVII coin:

[Head referee Jerry] Markbreit botched the coin toss during Super Bowl XVII. Dolphins captain Bob Kuechenberg called “tails,” and the coin came down “tails.” However, Markbreit became confused by the similar design of both sides of the coin: one side had two helmets and the other side showed two players holding helmets. Thus, he incorrectly thought “heads” had landed.

I challenge anyone to get ten heads in a row without a two-headed coin. In fact, this is the most likely correct solution to the original problem. A fallible human might in fact describe a two-headed coin as “fair” as long as both sides had an equal probability of showing. The right answer is 1, not 1 / 2^1,000,000. Our answer was off by about 100%.

Reporting the answer as 1 / 2^1,000,000 requires a little bit of statistics knowledge hiding in a giant vat of gullibility. What it says is that If an authority say that the coin is fair, no amount of evidence to the contrary will convince you otherwise. When reality collides with authority, you choose authority.

This has, I think, something to do with the barometer question:

“Describe how to determine the height of a skyscraper with a barometer.”

One student replied:

“You tie a long piece of string to the neck of the barometer, then lower the barometer from the roof of the skyscraper to the ground. The length of the string plus the length of the barometer will equal the height of the building.”

This highly original answer so incensed the examiner that the student was failed immediately. The student appealed on the grounds that his answer was indisputably correct, and the university appointed an independent arbiter to decide the case.

The arbiter judged that the answer was indeed correct, but did not display any noticeable knowledge of physics. To resolve the problem it was decided to call the student in and allow him six minutes in which to provide a verbal answer that showed at least a minimal familiarity with the basic principles of physics.

For five minutes the student sat in silence, forehead creased in thought. The arbiter reminded him that time was running out, to which the student replied that he had several extremely relevant answers, but couldn’t make up his mind which to use. On being advised to hurry up the student replied as follows:

“Firstly, you could take the barometer up to the roof of the skyscraper, drop it over the edge, and measure the time it takes to reach the ground. The height of the building can then be worked out from the formula H = 0.5g x t squared. But bad luck on the barometer.”

“Or if the sun is shining you could measure the height of the barometer, then set it on end and measure the length of its shadow. Then you measure the length of the skyscraper’s shadow, and thereafter it is a simple matter of proportional arithmetic to work out the height of the skyscraper.”

“But if you wanted to be highly scientific about it, you could tie a short piece of string to the barometer and swing it like a pendulum, first at ground level and then on the roof of the skyscraper. The height is worked out by the difference in the gravitational restoring force T =2 pi sqr root (l /g).”

“Or if the skyscraper has an outside emergency staircase, it would be easier to walk up it and mark off the height of the skyscraper in barometer lengths, then add them up.”

Unfortunately, I don’t think this ever actually happened, but every smart student has had some intentional or unintentional version of this experience at some time.

Photo credit: Columbia Pictures

The student in this anecdote makes the mistake of taking the question honestly. The building is a lie, as is the professor’s interest in its height. The student is Eva Mendes going off script and asking “Why are you pretending to not be Will Smith? You don’t even have a fake mustache.”

Conceits are necessary if you want to enjoy a movie or ask certain questions for instructional purposes. I do wonder, however, whether inflexible assessment questions have the effect of indoctrinating students to be unable to see through false conceits in real life. People seem to have difficulty distinguishing actors from the people they portray. Is it so hard to believe that a grown-up student would have a hard time distinguishing between information reported by a computer or provided by a boss and Truth? Will they be able to question the question?