Can't Touch This

A surprisingly common notion that comes up in science popularization is that nothing ever touches. This comes up things from youtube videos to the remake of Cosmos. As far as the scientific facts go, they're right. When you touch something, say when you pick up a ball, the electrons in the atoms of your fingers repel the electrons in the atoms of the ball, so the atoms in your fingers never come near the atoms in the ball. Near, that is, relative to the size of an atom.

But I wouldn't say that means nothing ever touches. Rather, it's a microscopic description of the macroscopic phenomenon of touch.

As an analogy, consider temperature. You can feel temperature as things feel hot or cold. You can measure temperature with thermometers. You can come up with laws that describe how heat flows from hot things to cold things. But on a microscopic scale, temperature is just speed. When atoms and molecules vibrate faster, they're hot. When they vibrate slower, they're cold.

But that doesn't mean that temperature doesn't exist. Rather, that's what temperature is. Thermometers still work, and the laws of thermodynamics are still accurate. They just refer to an emergent property of a complex system, rather than fundamental property.

I would argue the same applies to touch. Electrons repelling each other is what touching is.

Admittedly, this is entirely an argument over semantics. It's just about the definition of the word "touch", rather than any actual facts. But I think this definition is better and more useful. Because if nothing ever touches (except maybe where fusion occurs, like in the heart of a star), then the word "touch" never makes any useful distinctions, which is the purpose of a word.

The Monty Hall Problem

Suppose you're on a game show, and you're presented with three closed doors. Behind one of the doors is a car, and behind the other two are goats. You pick a door, let's say door 1. Before opening that door, the host, Monty Hall, opens a different door, let's say door 2, revealing a goat, and asks you if you want to switch which door you choose. Should you switch?

Counter-intuitively, the standard answer is yes. There is a 2/3 probability that the car is behind door 3, and only a 1/3 probability that the car is behind door 1. I'll prove it with math.

Bayes theorem says. In English, the probability of a hypothesis H given a piece of evidence E is equal to the probability of the evidence given the hypothesis times the prior probability of the hypothesis divided by the prior probability of the evidence

Let's define a few variables. We'll say C1, C2 and C3 stand for the car being behind door 1, 2 or 3, respectively. We'll also say O1, O2 and O3 stand for Monty opening door 1, 2 or 3, revealing a goat.

So, what we're interested in finding is P(C3|O2) or P(C1|O2). So, let's plug our variables into the equation.

Ok, so what are each of those terms? Well, P(O2|C3) = 1. Monty can't open the door you chose, and he's not going to open the door with the car. That only leaves one option. P(C3) = 1/3. With no information, we have to assume there's equal probability for the car to be behind each door.

By the law of total probability,, since Monty can choose either door 2 or door 3. P(O2|C2) = 0, since Monty won't open the door with the car behind it. Thus

Plugging those numbers back into the equation, we get. By the same logic,.

Here's another way to think about it. When you pick door 1, it either has the car behind it (probability 1/3) or it doesn't (probability 2/3). When Monty opens door 2, it doesn't change those probabilities. There's still a 1/3 probability your door has the car, and a 2/3 probability your door doesn't have the car. Since door 2 now has a probability 0 of having the car, that means door 3 must have the whole 2/3rds.

But what's really interesting to me about the Monty Hall problem is that it's not just dependent on what Monty does, it's also dependent on why he does it. Everything I've said is true for the standard problem, in which Monty always opens a door, and always reveals a goat. But if those things change, Monty can perform exactly the same actions, and get exactly the same results, but we'll still get different probabilities.

For example, suppose Monty only opens another door if you picked the door with the car. You pick door 1, and Monty opens door 2, revealing a goat. Should you switch? In that case, you definitely don't want to switch, because Monty wouldn't have opened a door at all if you had guessed incorrectly the first time. In this case, P(O2|C3) = 0, so P(C3|O2) = 0, and P(O2) = P(O2|C1)*P(C1), so P(C1|O2) = 1.

Alternatively, suppose Monty always opens a door, but opens one of the two you didn't pick at random. Again, you pick door 1, and Monty opens door 2, revealing a goat. It's possible Monty could have opened door 2 and revealed the car, but not this time. In this case, the probability that the car is behind door 1 and the probability that the car is behind door 3 are both 1/2. The reason is that he's twice as likely to open a goat door if you've chosen the car door than if you had not chosen the car door, so you do get information about the door you chose.

You can also find that using Bayes' theorem as before. P(O2|C2) = 0, since the goat can't be behind the door with the car. P(O2|C1) = P(O2|C3) = 1/2, since there's a 1/2 probability that Monty will open door 2, regardless of which door the car is behind.

But what if you don't know what strategy Monty is following? What if all you know is that there's a car behind one of the three doors, you picked door 1 and Monty opened door 2, revealing a goat. Maybe he's making a decision based on one of the three strategies I just described. Maybe he's using some other strategy. How do you calculate the probabilities then? What decision should you make?

I honestly have no idea.

A Defense of the Free Market

I've written before in defense of socialism, but that doesn't mean I'm opposed to the free market. Quite the opposite, in fact. The free market is a very powerful tool, and under the right conditions it's maximally efficient.

However, like any tool, it can be misused and abused. But that doesn't mean it should be completely banned, just that it should be used in a carefully controlled manner. There are those who think that regulating a market makes it non-free, but they're wrong. Regulations can help establish the conditions a free market needs to work efficiently.

One of the best uses for the free market is the production of luxuries, for example, video games. They're not a necessity, like medical care, so no will die if they can't afford them. They're not a natural monopoly, like electrical transmission, so lots of people can make them and drive competition. They don't have significant externalities, like pollution, so the costs and benefits are primarily borne by the buyers and sellers. And the free market environment is great for the spurring greater innovation and variety.

That is exactly the kind of thing for which the free market should be used.