These are real-world topics which (in more polite terms) attract more attention and argument than they deserve.
A separate page discusses fantasy obsessions.
On the mathematical side, the
Monty Hall problem
and Benford's Law are, in basic form,
simple and well explained in the Wikipedia pages, but attract unnecessarily convoluted explanations.
Inane media comparisons for the chance of winning the lottery
This is a pet personal peeve.
In early 2016 the Powerball lottery prize reached the record level of almost 1.6 billion dollars.
A single ticket has a 1 in 292 million chance of winning.
I often get contacted by journalists on such occasions, because there will be
many short news articles comparing this chance to a
list of asserted chances of other unlikely events.
The link above shows three such lists.
As well as the dubious accuracy of some figures there, and the fact that
those other chances depend very substantially on individual behavior,
the conceptual problem is that these other activities are hardly comparable to "buying one lottery ticket".
If one must seek comparisons, a sensible alternative is to standardize on a
one in a million chance.
Buying 6 Powerball tickets every week for a year adds up to about a 1 in a million chance
to win once.
To compare to another "ordinary" activity, this is similar to the chance of you being
killed in your next 200 mile automobile trip.
The mystique of card counting
The popular history of blackjack card counting
makes a great story, parts of which are actually true.
But like other adventure stories, it is not so advisable to try it yourself.
After quite considerable effort you might become competent enough to make a small profit,
and after more effort you might start to make a noticeable profit.
And then (in most places) be banned from casinos.
To me there are two take-away lessons here.
First, if an occupation in which you sit in an enclosed room and focus intensely
on a repetitive activity appeals to you, then more worthwhile occupations are available -- air traffic controller, for instance.
Second, I suspect that the casino industry
has actually benefitted from the existence of card counting raising the profile of blackjack amongst the general public
and thereby attracting more players; they have no reason to crack down on minor players.
The mystique of outstandingly successful investors
Logic says that, because most stock market investment is via professional management
(I'll write mutual funds, for simplicity) which charge a fee to investors, the average return to investors over all such funds
must be essentially the overall market average, minus the average fee.
Despite academics, and index fund groups such as Vanguard, emphasizing this fundamental and empirically
verifiable truth since the 1970s, it has taken a long time for passive investment to become commonplace.
One of several reasons is surely "wishful thinking" by investors who believe (or, more likely, are sold the belief)
that they can somehow pick a better-than-average fund.
This relates to the mystique of outstandingly successful investors, exemplified by
Peter Lynch in mutual funds,
Jim Simons in hedge funds,
Warren Buffett's holding company.
There are two issues in thinking about such people.
With tens of thousands of managers, some "by chance" will do better, and a few will be very lucky.
In order to judge whether the best performers did better than what could be expected by chance for the best,
one would need to know funds' risk profiles, and to take account of effects such as
I do not know any convincing analysis.
But for the sake of argument let us stipulate that indeed a handful of managers have indeed
been demonstrably better than explainable as luck.
That would be interesting as a fact about human ability.
But does it help you?
By analogy, let us imagine that there are a few struggling young artists whose work can be bought cheaply today
but will be very valuable in 20 years; how do we identify these particular artists?
There is a lot of data showing that, within the bulk of fund managers,
there is little correlation between performance in successive years, and simply looking at the best performers over the last
5 or 10 years is not enough to identify the special handful.
You can see for yourself by playing This online simulation game by my STAT 157
students Yixin Shen and Dodo Qian.
Hot hands -- phenomenon or fallacy?
An interest in extremes and records is a common human trait -- I imagine there is some
psychological research on why this is so -- and in the context of sports
people pay attention to streaks of successes of some type.
Alan Reifman's 2011 book
Hot Hand: The Statistics Behind Sports' Greatest Streaks
and his blog
contain many examples.
The "hot hand" phenomenon
originally referred to successive attempts by a player within a game
(shots in basketball being the prime example) being successful.
Many players believe that sometimes they are "in the zone" and for a period of time
perform better than usual.
If true, one could in principle detect it by looking at long streaks
and comparing observed lengths with what would be expected "just by chance"
for that player's skill level.
Now a priori this a reasonable question worthy of statistical study.
But there have been many studies, some of which find no evidence of this "hot hand" and others
find a small but statistically significant effect.
So a posteriori we may confidently conclude that the effect is at most small
and so the bottom line is
it doesn't matter whether it's real or not.