Implicit and explicit lists of representative examples of chance
Annotated list of contexts where we perceive chance
has several purposes.
On this page we take it as a descriptive reference point
for comparing other writers' focus and breadth to our own.
Elsewhere (xxx not yet written) we comment on using such lists as a
basis for taxonomic analyses.
below is the only loosely comparable explicit list I know.
Of course, by reading any author and seeing what examples they employ one can extract some
implicit short list of what they appear to view as representative examples.
Such lists comprise the remainder of this page.
This list, discussed in the opening chapter of
gives 20 instances intended to illustrate the author's view of the range of uncertainty.
As a Bayesian, he regards all instances of uncertainty as expressible in probability terms.
18 of his instances fit reasonably well with my own list, as follows
(here and below numbers refer to my list, as of July 2010, and I have re-ordered items according to my list order).
I would exclude from my list instances where it hard to visualize both alternatives
- (3) If an election were to be held tomorrow, 48% would vote Democrat.
- (3, 48) The proportion of HIV cases in the population currently exceeds 10%
- (5) A card drawn from a well-shuffled deck will be an ace.
- (6) The horse, High Street, will win the 2.30 race.
- (10,15) The flight will arrive in London tomorrow morning.
- (22,81) Shares in pharmaceutical companies will rise over the next month.
- (25) There will be a serious nuclear accident in Britain next year.
- (26) It will rain tomorrow.
- (35) Inflation next month will be 3.7%
- (35) The planting of genetically modified crops will damage the environment.
- (44) The defendant is [truly] guilty.
- (45) The addition of selenium to your diet will reduce your chance of getting cancer.
- (45) The British should reduce the amount of saturated fat in their diet.
- (46) The Princes in the Tower were murdered on the orders of Richard III.
- (46) Many eighteenth century painters used lenses and mirrors.
- (46) Mrs Anderson was Anastasia, daughter of the last Tsar of Russia.
- (47) The skull is 7 million years old and is that of a hominid.
- (48) The capital of Liberia is Monrovia.
At first sight this list has a pretty broad range.
At second sight, the first half tend toward familiar textbook examples
of probability/statistics, and the second half has a tight focus on the
correctness of scientific/historical theories.
Alas these examples are not actually discussed seriously in the rest of the book.
- Jesus was the son of God.
- The sun will rise tomorrow at the time stated.
In many ways, the book that best complements our list is
Nate Silver's The Signal and the Noise.
Even though restricting itself to predictions about the future
(rather than all of chance and uncertainty) the 13 chapters of
this book mostly correspond closely to contexts on our list.
The book gives an interesting chapter of discussion of each topic, with a nice combination of
details and overview.
The author has deliberately chosen
contexts where there is a lot of past data, and the central issue (his signal/noise analogy) is determining which
aspects of the data are useful in predicting the future.
- (3) accuracy of opinion polls vs expert assessments.
- (6) sports betting.
- (7) baseball player's performance.
- (8) professional poker.
- (16) flu pandemics.
- (26) weather.
- (32) mortgage default likelihoods.
- (34) predicting business cycle/economic indicators.
- (36) terrorism.
- (45) climate change.
- (65) predicting earthquakes.
- (81) stock market, efficient market hypothesis, bubbles.
- (84) Herding, overconfidence.
Life of Norm
Of the 27 short chapters in Michael Blastland and David Spiegelhalter's
The Norm Chronicles: Stories and Numbers About Danger, 22 are on
quite specific topics (rather than chance in general) and aside from the rather broad "life expectancy"
chapter, they fit well into our categorization.
(15) accidents to children
(15) accidents involving transportation
(15) workplace accidents
(15) accidents involving extreme sports
(16) health effects of diet/exercise/smoking/alcohol
(16) medical risks to child in birth and infancy
(16,19) sex - disease and accidental pregnancy
(16,17) risks from use of illegal drugs or abuse of prescription drugs
(18) screening for disease
(19) risks to mother in giving birth
(20) violence/abduction to children
(20) crime to adults
(24) medical expenses after retirement
(45) climate change
An Introduction to Probability and Inductive Logic,
the brief chapter on philosophical interpretations of probability
concludes with the following comments
(typical of many other discussions of the frequentist/Bayes philosophies).
Our prototypical examples [of probability] are artificial randomizers.
But as we start to think hard about more real-life examples, we get further and further away from the core examples.
Then our examples tend to cluster into belief-type examples, and frequency-type examples, and in the end we develop ideas
of two different kinds of probability.
The text is accompanied by a graphic, in which the following 13 "examples" are arranged in a circle around a central entry
- (5) Urns
- (5) Lotteries
- (5) Cards
- (5) Dice
- (9) Coincidences
- (15) Frequency of traffic accidents
- (26) The weather
- (41) Radioactive decay
- (44) Guilt of accused criminals
- (51) The single case
- (61) Probability of a live birth being female
- (74) Telephone waiting times
- (96) Extinction of the dinosaurs
(xxx move to a taxonomy page).
Short lists of examples are appropriate and indispensible for illustrating
a distinction implied by a definition (e.g. qualitative vs quantitative variable)
or a distinction that is uncontroversially substantive and useful
(e.g. marine mammal vs fish).
But if you wish to put forward an argument that some distinction is
substantive and useful then a short list of iconic examples on both sides
is not at all convincing.
You need to xxx show that "most" examples can be decisively put on one side
or the other, and that xxx not too unbalanced.
xxx need long list
xxx need list not chosen by you!
von Mises examples
von Mises gives, in Probability, Statistics and Truth,
a "summary of his theory in sixteen propositions", and here are the first three.
So this is the "dogmatic frequentist" position:
- The statements of the theory of probability cannot be understood correctly if
the word `probability' is used in the meaning of everyday speech; they hold only for
a definite, artificially limited rational concept of probability.
- This rational concept of probability acquires a precise meaning only if the
collective to which it is applied is defined exactly in each case. A collective is a
mass phenomenon or repetitive event that satisfies certain conditions; generally
speaking, it consists of a sequence of observations which can be continued
- The probability of an attribute (a result of observations) within a collective
is the limiting value of the relative frequency with which this attribute recurs in
the indefinitely prolonged sequence of observations. This limiting value is not
affected by any place selection applied to the sequence.
He explicitly excludes ``everyday" examples and e.g. weather forecasts.
Here is what remains, as his in text examples.
numerical probabilities only make sense in the context of repeatable chance
- (4) Drug tests
- (5) Dice,
- (5) Coin tosses,
- (5) Lotteries,
- (5) Urns
- (23) Life expectancy,
- (23) Life insurance
- (41) Quantum theory,
- (41) Radioactivity
- (53) Shooting at a target (as prototype of repeated chance experiment)
- (54) The "theory of errors"
- (61) Population genetics,
- (61) Sex ratio at birth
- (64) Statistical physics,
- (64) Brownian motion.
An interesting implicit list from the philosopher Antony Eagle appears in his article
Randomness Is Unpredictability (specifically
section 1, Randomness in science ).
Many items are indicated by only brief in-sentence phrases, quoted below.
Covers a broader range than many philosophers do.
The article's premise -- that "Randomness Is Unpredictability" is a rather novel
philosophical idea that needs justification -- seems bizarre to a statistician or scientist.
To the question "what does it mean to say that the result
of a die roll is random?", surely the most common answer "it's random in the sense of
a very standard topic (our (54): Residuals (errors) in estimation) is that the errors
in your best prediction must be random in a certain sense, otherwise you could improve the prediction.
- (3) In many statistical contexts, experimenters have to select a representative
sample of a population
- (4) In Fisher's famous thought experiment, we suppose a woman claims to be
able to taste whether milk was added to the empty cup or to the tea
- (5) gaming devices such as coins and dice
- (8) Another good example of random behaviour occurs in game theory.
- (41) quantum mechanical systems.
- (61) the Hardy-Weinberg Law
- (62) A further example is provided by the concept of random mutation in classical
- (65) Examples from "chaos theory" have been particularly prominent recently
- (72) Therefore, these models include a random noise factor: random alterations
of the signal with a certain probability distribution.
- (87) processes that are modelled by probabilistic state transitions ... [e.g.]
the way that present and future states of the weather are related
- (???) randomness of the rainfall input is important in explaining the
robust structure of the dynamics of soil moisture.
The examples in the four books above were rather easy
to fit into our contexts. I suspect this is because
the authors actually started by thinking of a "context" then invented an
example. When you take actual specific real-world examples
(like the final one above)
it becomes harder to fit into prespecified contexts. Unsurprisingly!
As a representative of ``popular science" style books on Probability,
let us take Leonard Mlodinow's
Drunkard's Walk: How Randomness Rules Our Lives.
- (3) Opinion polls
- (5) Gambling; lotteries, cards, dice, roulette
- (7) Baseball statistics, hot hands
- (14) Success in individual careers (Bill Gates etc)
- (18) False positives in medical tests
- (23) Life expectancy, insurance
- (25) Unpredictable reliability of human artifacts (Three Mile Island)
- (35) Unpredictability in geopolitics (Pearl Harbor)
- (44) DNA profiling, reliability of eyewitnesses
- (54) Measurement error
- (64) Brownian motion
- (83) Poor intuition about chance (probability matching, imputing pattern to randomness)
2015 New Scientist erxamples
The 12 March 2015 New Scientist cover emphasizes their special report
(14 pages; 3 authors plus 6 interview columns) entitled
Chance: how randomness rules our world.
Their examples are
Analysis. This has a nice broad range, but the specific topics chosen seem very conventional and could mostly been described similarly 30 years ago.
The only imaginative topic here is avalanche prediction.
- (5) poker
- (6) sports gambling
- (9) coincidences and rare events
- (26) weather forecasting
- (26) avalanche prediction
- (41) quantum theory
- (44) statistics in law
- (55) frequentists vs Bayesians
- (62) chance mutations
- (65) deterministic chaos
- (73) random number generators
- (93) evolution of complex life
xxx Rosenthal lightning book.