(i) Probability distributions on continuous variables
(and on integers, if the spread is large) may be assumed to be "smooth",
unless there is some specific reason why they may not be.
(ii) In the setting of (i), quantitative predictions for data based on the
assumption that the data
arises from independent random picks from some unknown "smooth"
distribution are usually approximately correct, even when the
``independent random picks" part of the assumption is unrealistic.
As usual, let me distinguish the philosophical issue -- why do we think the assumptions in (i) or the predictions in (ii) are reasonable? -- from the empirical issue of when they work out to be give correct predictions.
Note that the usual math setup "builds in" assumption (i) for continuous variables as a default by working with density functions -- unless one is explicitly modeling e.g. a fractal -- but that doesn't address either issue.
A darts player aims at the center of the bull's eye (which has diameter 0.50in); the distributions of horizontal and of vertical deviations have mean 0 and s.d. 1.50in. Assuming these deviation have independent Normal distributions, what is the chance of hitting the bull's eye ?
The intended answer is "about 1.4%". The model is not very realistic (one expects smaller horizontal than vertical s.d., and the Normal distribution is doubtful) but setting aside the specific distributional model, the conceptual point is that the "local uniformity principle" says we may assume there is a smooth density function f, and then we estimate the desired chance in terms of f, to a first approximation as area(bull's eye) x f(0) where f(0) is the density at the bull's eye.
xxx coin tossing
seems intuitively reasonable but why?
There are several ways of estimating the chances of an asteroid collision
with Earth.
The actual frequency of different sized asteroids at different points in
the solar system is of course an empirical issue.
But because the (quarter million mile) distance to the Moon is small
relative to the solar system,
the "local uniformity principle" says we may assume
that the chance of an asteroid being at one point near the Earth's orbit
is not substantially different from the chance of it being at a different
point a quarter million miles away.
Then mathematics, and the empirical fact that the ratio (radius of Moon
orbit)/(radius of Earth) is approximately 60, implies
Amongst asteroids which pass closer to earth than the Moon's orbit, about
one in 3,600 will hit Earth.
(the 3600 arises as 60^2).
Hypothetically, if astronomers could and did detect all asteroids of
diameter greater than 50 meters passing within the Moon's orbit for a
period of years, and found there were on average 3.6 per year, then one
could infer that such an asteroid would hit the Earth about once every
1,000 years on average.
(Note: this conclusion is a
typical estimate
in the literature but was not obtained in this way.)
xxx exam scores
xxx last digit of travel expenses 100-1000 dollars
xxx Benford's law
xxx empirical: diff sources mix -- not The Secret Life of Numbers