Some people may have had this experience: after working out a probability problem, the answer seems unbelievable — how can the probability be so high, or so low? Math majors, when approximating with different methods, sometimes get results that differ by several times over, which breeds suspicion (see p.322, problem 15). To put the question another way: is probability reliable? Is it real?

Here’s a fairly famous example in probability (p.23), put in terms we’re familiar with: if a class has 40 people, the probability that no two of them share a birthday is 0.109. The larger the class, the lower this probability drops — by 55 people it’s down to just 0.01. Think about all the classes you’ve joined over the years — surprising, isn’t it? It shows that probability isn’t always so useful. There are many more examples like this, such as the de Méré problem in probability (p.30).

This is the kind of answer you get from calculating something theoretically. Theory is like an abyss — toss a pebble in and it’s like a tiger borrowing a pig: things go in but never come out. You can imagine that flipping a coin a few thousand times and getting heads every time wouldn’t be such a big deal either. And even setting aside the purely scientific angle, probability is, to some extent, useless.

When using Bayes’ formula there’s the issue of a prior probability (p.64), and that probability is subjectively assigned. Once subjective factors get mixed in, many things change — we automatically delete certain things we don’t want to see, which in effect inflates the probability of certain other things happening. People really do this kind of foolish thing all the time. Take the use of fabricated probabilities in academia — in early psychology research, some people manufactured data like this to “prove” the correctness of their own claims. We always say such people are “not rigorous,” but as mentioned above, rigor isn’t necessarily useful either.

Here’s an example: the probability that we’ll die the moment we open the door in the morning is 50%. But factoring in the chance of a building collapsing, a car speeding out of control, someone’s poorly controlled pet biting you to death… the probability that opening the door kills you would be far higher than 50%. From this, it seems most of us are quite lucky. But when calculating this probability, we subjectively consider only the lethal factors and not the non-lethal ones — if we also factored in the non-lethal factors (the “harmonious society” factor), the probability of dying might become very, very low. So which is the scientific one — the objective 50%, or the subjective non-50%? Looking deeper, in considering whether stepping outside kills you or not, we’re performing something like a “you (not dying)–me (dying)” shift of perspective — should we think with the subjective eye or the objective eye? Does probability belong to theory, or to reality? Seen this way, a lot of what’s in probability really does resemble philosophy — except it’s the damned feelings and data hysterically fabricated by humanity, just there to fool you.

Who really knows? Maybe Bernoulli does.

Reference: Li Xianping, Foundations of Probability Theory, 2nd edition, Higher Education Press