FIVE- M IN U T E A N A LYST
SOME EXCURSIONS
Suppose I could have one piece of
information. A likely choice would be:
“How much better off would we be if
we knew the eventual winner?” In this
case, we would reduce the number of
possibilities in each round by 1; and the
number of combinations would “only” be
, which
is 64 times better than the original estimate. Sixty-four is generally reckoned
to be a small number when compared
18
to 10 . If you knew the Final Four, you
would be considerably better off, at
,
or 520,000 times better than the original
bet, which is to say that in the scheme
of things, you are no better off at all.
Now suppose that you had to pay
$1 to play this game instead of it being
free, but you think you are pretty good at
predicting basketball games. You would
need to have 72 percent accuracy in
your ability to pick basketball games to
be risk neutral for a dollar (i.e., 72 percent accuracy increases your odds of
winning to 1 in 1 billion).
As bad as these odds are, here’s a
game that is even worse, which I will
call the Georgetown Wager (after my
colleague who challenged me to come
up with a tougher game). Suppose that
you are given the 64 teams that will play,
but the games are randomized and you
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have no information about their parings;
you know that there are 64 teams in the
first round, of which 32 will win, and so
on, but you don’t know who will play
who. In this version, you have to figure
out for the number of possibilities,
(1)
where the “choose” function,
if you expand
(1) out by hand and cancel terms4, you
will find that it is:
.
If the odds of Buffet’s Billion Bracket
are bad, the Georgetown Wager is patently absurd; the odds of winning are
, which are roughly on the
order of winning Buffet’s game twice in
a row!
You may be asking: “So, if this is
such a good bet for the house, why
don’t I run a similar lottery?” Because
I don’t have a billion dollars, and I’m
not willing to lose. Remember that the
“house” has to be willing to pay out the
fee in the extraordinarily rare event that
someone won. Events that are “statistically impossible” are still “possible.”
and while it is extraordinarily unlikely
that someone will win, there is no law
of physics that prevents someone from
winning.
W W W. I N F O R M S . O R G