Alternate
Bets with Equal Edge May Perform Differently.
Picture a game like roulette
but with 34 outcomes -- 0, 00, and 1 through
32. And think about two types of bets. Straight-up,
on single numbers, paying 31-to-1; the likelihood
of winning is 2.94 percent (one out of 34).
Outside, for instance odd values between 1 and
32, paying 1-to-1 and losing on 0 or 00 as well
as on even numbers; prospects of winning are
47.0 percent (16 out of 34). The house advantage
works out to be 5.88 percent in either case.
Picture
a game like roulette but with 34 outcomes --
0, 00, and 1 through 32. And think about two
types of bets. Straight-up, on single numbers,
paying 31-to-1; the likelihood of winning is
2.94 percent (one out of 34). Outside, for instance
odd values between 1 and 32, paying 1-to-1 and
losing on 0 or 00 as well as on even numbers;
prospects of winning are 47.0 percent (16 out
of 34). The house advantage works out to be
5.88 percent in either case.
Fantasize
that your favorite casino will book action down
to $1 anywhere on the table. More, say you're
a real sport and decide to risk $1, setting
a goal of earning $31 or forfeiting your buck.
And, make believe time is unimportant you'd
as soon grind it out over a long session or
do it in one fell swoop.
You
accordingly devise three alternatives. There
are other possibilities, but these seem to offer
some good options. The first is to bet the dollar
on a single spot, succeeding or failing on one
spin. The second is to start with your money
on Odd and, if you get lucky, parlaying your
earnings; that is, plowing it all back up to
five times, balancing profits of $1 + $2 + $4
+ $8 + $16 = $31 against a wipe-out. The third
is to grind away at a dollar a pop. Here, you
can hit your goal in as few as 31 successive
wins. Or, it may take a run of 32 wins and one
loss, 33 wins and two losses, and so on -- providing
that the losses don't knock you out before the
wins get you to the top.
You'd
start with the same dollar in each instance,
looking for an identical profit, and making
bets all with equal edge. Would your hopes for
prevailing be equivalent as well, such that
the only distinction would be the time factor?
The
probability of victory betting one spin on a
single number is simply the 2.94 percent or
one out of 34. The parlay needs five wins on
Odd in a row. That's 47 percent multiplied by
itself five times, which comes out to 2.31 percent,
just under one out of 43.
Betting
a dollar per spin, you could amass the $31 by
winning 31 rounds in a row. This is 47 percent
multiplied by itself 31 times, or one out of
less than 14 billion. Reaching $31 with this
strategy through combinations of wins and losses
greatly improves the chance of triumph, to roughly
0.3 percent or one out of 335.
The
varying outlook for reaching the $31 profit
starting with $1 partly results from the fact
that the handle, the total amount wagered, differs
among the methods. And the edge acts to shift
a fraction of the handle from the solid citizens
to the bosses. The penalty imposed by the edge
on players therefore rises as the gross wager
increases.
At
5.88 percent edge, $1 on a single number, one
time, gives the house a theoretical cut of 5.88
cents. The parlay involves $1 on the first round
-- 5.88 cents for the house; then $2 -- 11.76
cents for the house, followed by $4 at 23.52
cents, $8 at 47.04 cents, and $16 at 94.08 cents.
The total is over $1.82. Betting $1 on each
of 31 spins would also cost $1.82 in commissions
to the casino. Improving the chance of winning
by allowing some losses and greater numbers
of wins would raise this amount. For instance,
a series including 9 losses and 40 wins, a total
of 49 spins, represents a theoretical $2.88
for the house.
There's another difference. Your goal might
be to go all the way to $31 or bust out trying.
Putting the dollar on a single number and taking
one shot, there's no turning back. The parlay
offers four occasions to change your mind and
quit with an intermediate profit, but a single
loss still sends you to the lockers. And betting
$1 per spin offers multiple opportunities to
skip with the joint's jack, along a broader
range of stopping points; and depending where
you are in the sequence, a loss or two doesn't
necessarily spell curtains.
Had
Aesop considered such trade-offs, the fable
of the Tortoise and the Hare would have been
far more complex than it is. For, as the sagacious
scribbler, Sumner A Ingmark, wryly wrote:
Among
a pragmatist's contentions:
Reality has more dimensions,
Than most philosophers' inventions.
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