# Probability is subtle

When I was in college, I overheard two senior faculty arguing over an undergraduate probability homework assignment. This seemed very strange. It occurred to me that I’d never seen faculty argue over something elementary before, and I couldn’t imagine an argument over, say, a calculus homework problem. Professors might forget how to do a calculus problem, or make a mistake in a calculation, but you wouldn’t see two professors defending incompatible solutions.

Intuitive discussions of probability are very likely to be wrong. Experts know this. They’ll say things like “I imagine the answer is around this, but I’d have to go through the calculations to be sure.” Probability is not like physics where you can usually get within an order of magnitude of a correct answer without formal calculation. Probabilistic intuition doesn’t take you as far as physical intuition.

## 12 thoughts on “Probability is subtle”

1. mark leeds

Hi John: The Monty Hall problem and the birthday problem are
two good examples of that.

2. John:

I disagree! I was a physics major in college. I did very well in my classes but I decided to stop studying physics because I felt my physical intuition was terrible. In contrast, my probabilistic and statistical intuition is pretty good! So I think it depends a lot on the person.

3. the cyclist

You’re probably right.

4. GlennF

Hi John,

Can you share what problem they were arguing over?

5. Unfortunately I don’t remember the problem.

6. Imre Koncz

Good point. Moreover, the most talented geniuses had been terrible wrong, in elementary problems!
An example: throwing two fair coins, what is the probability that there was at least one head?
Leibniz and d’Alambert both answered 2/3! (the right answer is 3/4, “of course”), they thought that there are 3 cases HH,HT,TT and two of them is good for us. This opinion was even printed in 1975 the in Encylopedie, as “Croix ou pile”…
I can hardly imagine to make such a mistake in physics, or calculus…

My other note: I used to teach high school and university students.
Analysis or linear algebra was full of joy, i could assist to get to the right answer almost everybody, most of the times i could give good analogy and intuition. In worst case, the student could learn and apply the univocal and straightforward rules, and could derive or calculate what they should. But teaching prob theory is a real nightmare. Problems like: what is the prob to get two aces in a poker game (5 cards in hands) proved to be extremely hard to explain, or even to convince the student about the right answer.
One of the root of the problems is what is the elementary event. (calculate prob = good events/all events ). It is not straightforward, not obvious at all!
To point back to my first point, even d’Alambert “failed”.

The situation is even worse: i realized that i cannot convince a student that HH and HT are not the same prob elementary events. They asks Why? Just. Just because the teacher says :)
So one can realize, that the key is that we can distinguish between the two coins. So let paint one red, the other green.
Now think it again. Red head and green tail should be a  different event that the green head and red tail, whiles two heads is just one case. So far so good.

But wait. Painting the coins (or dices), whats going on. Why the hell color matters in a math problem? So, to tell the truth, there is no way to decide between the 3/4 and 2/3 solution just by using pure logic. Somehow, it is the physical properties of the coins, that they follow the “3/4” statistics. There are real object of the outer world, which cannot be distinguished from each other, hence following other statistics. E.g. electrons follows the so called bose-einsten statistics, and you cannot discriminate between them, while dices, coins follows Maxwell-Boltzmann statistics. This feature of electron has serious implication in physics, e.g. determines the symmetry properties of the wave function…
Actually, it is an experimental fact, that coins follows the statistics which resulting in 3/4. But it is not a result of a derivation! So, this problem is a physical problem, not a math problem, OR we should define first, what kind of statistics the dices are following….. uu..

OMG! This is the most elementary problem, like solved in the very first class, and now we are talking about quantum mechanics!
Honestly, before giving so hard problems to kids, we should alert them, that its not a math problem. It is math, only if we declare the elementary events and their probabilities as axioms, and then one should apply the Kolmogorov axioms.

Well, its all confusing… Anyhow, this can play a role in why probability is so hard, and why our intuition does not work.

7. I think a lot of confusion arises because the boundaries between mathematical probability theory and physical experiments are not always clearly indicated. (For instance, in the coin example that Imre Koncz discusses, the students may get confused because of this.)
In high school, my teacher explained about the impossible event and gave the example of a die landing on a vertex. This puzzled me. But now I know that it was not just a matter of it being physically highly unlikely for this to happen, but rather that we had defined the possible outcomes to be {1,2,3,4,5,6}. Since landing on a vertex is not represented in this set, all we can assign this ‘outcome’ to is the empty set, which is called the impossible event.
So, what wasn’t clear to me then is that what counts as impossible depends on the model. We could have modeled reality in a different way, with a richer set of outcomes, and then “land on a vertex” could have been a possible outcome.
So our physical intuition may actually hamper our probabilistic intuition if model and reality are confused in this way.

8. Sylvia’s comment reminds me of a “Physics of Basketball” lecture I heard in grad school.
Q: When does a basketball go through a hoop?
A: When the basketball is smaller than the hoop.
Tilt a circular hoop. watch the minor axis of the ellipse narrow and the geometry is clear. But, what about when the ball hits the hoop (or vertex in Sylvia’s example)? The professor’s simple, non-dissipative model fails. At that point, I tried to imagine the shape of the scoring region in phase space and lost the rest of the lecture… If he had just told us the boundaries between physical reality and his mathematical axioms/assumption, perhaps I would have paid better attention in lecture.

Re Imre’s comment about indistinguishable electrons, that’s not strictly true. You can distinguish between them in certain cases. An electron in a pi orbital is very different than one in a sigma orbital on the same molecule. They will respond differently to laser light so you can easily tell them apart.

What puzzled me, until a kindly professor set me right, was how to distinguish between the 3 oxygen atoms in ozone when they are in the triangular transition state. You can’t take the oxygen out of context (the field exerted by surroundings). That field stores a lot of information.

9. froginthewell

Were the senior faculty actually senior math faculty?

10. Yes. They were both functional analysts.

11. You know, this post is only two paragraphs, but it’s had a lasting impact on me because it pops in my head every once in a while. It’s a very interesting thought.

12. Thanks!