If I roll 15 fair 6-sided dice, take the ones that rolled 4 or higher, roll them again, and sum up all the die rolls... what is the probability that I drop at least one die on the floor?
There are two different ways of using probability. When we think of probability, we normally think of neat statistics problems where you start with numbers, do some math, and end with a number. After all, if we don't have any numbers to start with, we can't use a proven formula from a textbook; and if we don't use a proven formula from a textbook, our answer can't be right, can it? But there's another way of using probability that's more general: a probability is just an estimate, produced by the best means available, even if that's a guess produced by mere intuition. To distinguish these two types, let's call the former kind *formal probabilities*, and the latter kind *informal probabilities*.
An informal probability summarizes your state of knowledge, no matter how much or how little knowledge that is. You can make an informal probability in a second, based on your present level of confidence, or spend time making it more precise by looking for details, anchors, reference classes. It is perfectly valid to assign probabilities to things you don't have numbers for, to things you're completely ignorant about, to things that are too complex for you to model, and to things that are poorly defined or underspecified. Giving a probability estimate does not require *any* minimum amount of thought, evidence, or calculation. Giving an informal probability is not a claim that any relevant mathematical calculation has been done, nor that any calculation is even possible.
I present here the case for assigning informal probabilities, as often as is practical. If any statement crosses your mind that seems especially important, you should put a number on it. Routinely putting probabilities on things has significant benefits, even if they aren't very accurate, even if you don't use them in calculations, and even if you don't share them. The process of assigning probabilities to things tends to prompt useful observations and clarify thinking; it eases the transition into formal calculation when you discover you need it, and provides a sanity check on formal probabilities; having used probabilities makes it easier to diagnose mistakes later; and using probabilities lets you quantify, not just confidence, but also the strength and usefulness of pieces of evidence, and the expected value of avenues of investigation. Finally, practice at generating probabilities makes you better at it.
The first thing to notice is that informal probabilities are much more broadly applicable than formal probabilities are. A formal probability requires more information and more work; in particular, you need to start with relevant numbers; but for most routine questions, you just don't have that data and it wouldn't be worth gathering anyways. For example, it's worth estimating the informal probability that you'll like a dish before ordering it at a restaurant, but producing a formal probability would require a taste test, which is far outside the realm of practicality.
Assigning informal probabilities clarifies thinking, by forcing you to ask [the fundamental question](http://lesswrong.com/lw/24c/the_fundamental_question/): What do I believe, and why do I believe it? Sometimes, the reason turns out not to be very good, and you ought to assign a low probability. That's important to notice. Sometimes the reason is solid, but tracking it down leads you to something else that's important. That's good, too. Coming up with probabilities also pushes you to look for reference classes and examples. You can still ask these things without using probability, but trying to produce a probability gives guidance and motivation that greatly increases the chance that you'll actually remember to ask these questions when you need to. Informal probabilities also ease the transition into formal calculation when you need it; you can fill in an expected-utility calculation or other formula with estimates, then look for better numbers if the decision is close enough.
Probabilities are easier to remember than informal notions of confidence. This is important if you catch a mistake and need to go back and figure out where you went wrong; you want to be able to point to a specific thought you had and say, "this was wrong in light of the evidence I had at the time", or "I should've updated this when I found out X". Unfortunately, memories of degrees of confidence tend to come back badly distorted, unless they're crystallized somehow. Worse, they tend to come back consistently biased towards whatever would be judged correct now, which makes them useless or worse. Numbers crystallize those memories, making them usable and enabling you to retrace steps
Quantifying confidence also enables us to quantify the strength of evidence - that is, how much a piece of information *changes* our confidence. For example, a piece of evidence that changes our probability estimate from 0.2 to 0.8 is a likelihood ratio of 4:1, or 2 bits of evidence. Assigning before-and-after-evidence probabilities to a statement forces you to consider just how good a piece of evidence it is; and this makes certain mistakes less likely. It's less tempting to round weak arguments off to zero, or to respond emotionally to an argument without judging its actual significance, if you're in the habit of putting numbers on that significance. But keep on mind that there is not one true value for the strength of a piece of evidence; it depends what you already know. For example, an argument that's a duplicate of one you've already updated on has no value at all
Finally, assigning probabilities to things is a skill like any other, which means it improves with practice. Estimating probabilities and writing them down enables us to calibrate our intuitions. Even if you don't write anything down, just noticing every time you put a .99 on something that turns out to be false is a big improvement over no calibration at all.
I know of only one caveat: You shouldn't share every probability you produce, unless you're very clear about where it came from. People who're used to only seeing formal probabilities may assume that you have more information than you really do, or that you're trying to misrepresent the information you have.
To help overcome any internal resistance to giving informal probabilities, I have here a list of probability Fermi problems. A Fermi problem asks for only a rough estimate - an order of magnitude - and it does not include enough information for a precise answer. So too with these problems, which contain just enough information for an estimate. Answer quickly (ten seconds per question at most). Don't do any calculations except very simple ones in your head. Don't worry about all the missing details that could affect the answer. The goal is to be quick, since speed is the main obstacle to using probability routinely.
1. A car is white.
2. A car is a white, ten year old Ford with a dent on the rear right door
3. A ten-mile car trip will involve a collision.
4. A building is residential.
5. A person is below the age of 20.
6. A word in a book contaains a typo.
7. Your arm will spontaneously transform into a blue tentacle today.
8. A purse contains exactly 71 coins.
9. 76297 is a prime number.