Horse-race betting typically follows a pari-mutuel system. The term...
The "objective probability" is determined empirically. Since the tr...
Elizabeth Becknell's 1940 paper argued that human decisions are oft...
In 1937, Zenith Radio Corporation sponsored a nationwide radio expe...
Preston and Baratta (1948) ran laboratory experiments where subject...
The late 1940s marked the peak of horse racing's golden age in Amer...
Pari-mutuel (French for "mutual betting") means bettors wager aga...
In standard probability theory, odds are defined as the reciprocal ...
Before betting opened, track handicappers posted "morning line" odd...
**The Theoretical Minimum (7.8)**
The value **7.8** serves as a ...
Griffith's test works as follows: if bettors correctly assess proba...
ODDS
ADJUSTMENTS
BY
AMERICAN
HORSE-RACE BETTORS
By
R.
M.
GRIFFITH,
University
of
Kentucky
In
horse-race
betting,
the
odds
on
the
various
horses
in
any
race
are
a
functioning
of
the
proportion
of the
total
money
that
is
bet
on
each
and
hence
are
socially
determined.
On
the
other
hand,
the
objective
probability
for winners
from
any group
of
horses
is
given
a
posteriori
by
the
per-
centage
of winners.
Thus
the
odds
express
(reciprocally)
a
psychological
probability
while
the
percentage
of
winners
at
any
odds
group
measures
the
true
probability;
any
consistent
discrepancy
between
the two
may
cast
light
not
only
on the
specific
topics
of horse-race
betting
and
gambling
but
on the more
general
field of
the
psychology
of
probabilities.
Becknell
noted
with
her
idealogies
of
probability
that
in
many
situations
not
the
probability
but
the
judgment
of
the
probability
is the critical
variable.1
Goodfellow2
and
Fernberger3
have
experimentally
shown that such
ideologies may
influence
ESP
results. More
recently
Preston
and
Baratta
have demonstrated
a
systematized
under-
valuation
of
large probabilities
and an
overvaluation
of
small,
with
the indifference-
point
falling
near
the
geometric
mean of
their series
in
an
experimental
situation
and
have discussed
certain
parallels
in
adaptation-levels.4
This
study may
help
to
determine
the
universality
of their effect
by
submitting
it
to
a
complex,
non-laboratory
test.
Betting
on horse-races
offers
per
se
specific
but
non-trivial
psychological
problems.
Undoubtedly
the
most common
form
of
gambling,
it has
assumed
such
proportions
as
to be a
major
socio-economic
problem
and to constitute
large portions
of the
revenue of
condoning
states.'
Gambling
has been
analyzed
for
knowledge
of
psycho-
logical
and
economic
processes,6
and
pathological
gambling
has been
clinically
studied.7
The
emotionally
rich
background
of
fate
and
money
makes
this a
promising
field
for
observing
variety
in
personality.
*
Accepted
for
publication
October
5,
1948.
The author is indebted
to
Mr.
J.
A.
Estes,
editor
of
The
Blood
Horse,
for
source
material and
valuable
suggestions.
1E.
A.
Becknell,
Probability:
A
function
of
ideology,
this
JOURNAL,
53,
1940,
604-609.
2
L.
D.
Goodfellow,
A
psychological interpretation
of
the
results
of the Zenith
radio
experiments
in
telepathy,
J.
Exper.
Psychol.,
23,
1938,
601-632.
3S. W.
Fernberger,
'Extra-sensory
perceptions'
or
instructions?,
ibid.,
22, 1938,
602-607.
4M. G. Preston and
P.
Baratta,
An
experimental
study
of the auction-value
of
an uncertain
outcome,
this
JOURNAL,
61,
1948,
183-193.
'See
W.
Steigleman, Horseracing,
1947,
27.
In
1946
over
$1,700,000,000
were
bet
in the
pari-mutuel
machines. Of this
amount,
nineteen states
retained
$80,000,000.
Several times this
amount is bet
with
illegal
bookies.
6
E.
M.
Riddle,
Aggressive
behavior
in
a
small social
group,
Arch.
Psychol.,
12,
1925
(no 78),
1-196;
J.
von
Neuman
and
O.
Morgenstern,
Theory
of
Games
and
Economic
Behavior, 1944,
1-641.
O.
Fenichel,
The
Psychoanalytic
Theory of
Neurosis, 1945,
372
f.
290
ODDS
ADJUSTMENTS
BY
BETTORS
In
any
investigation
of
gambling,
insight
and direction
may
be
gained
from
such clinical
and
descriptive
works
as
The
Gambler
by
Dostoevski.
The briefest
perusal
reveals that not
all forms of
gambling
are
psycho-
logically
equivalent.
Games of chance
may
be
placed
in
a
hierarchy
of
skills.
The conative
predictions
of
the
fall
of dice contrast
with
the
cognitive
factors involved in
skillful
poker-playing. Many
people
bet on
horse-races because
of
an-be it as it
may-uncritical,
egotistical
ration-
ality
as
to
their
ability
to
select winners
through
handicapping
procedures.
In
so
far
as a
cognitive
function
is
involved
in
any
situation
one
may
question
the
psychological
applicability
of
the term
'gambling.'
The
most
tentative list of
ideologies
involved
in
horse-race
betting
would,
however,
touch
upon
fundamental
philosophies
of
probability
and
mathe-
matical
expectation,
and few of the
systems
of
betting
neglect
to
give
some
consideration to the
judgment
of
the
odds of
the
selections;
some
employ
them
exclusively,
as,
for
example,
when
a
system
permits
only
the
favorites
at
odds
exceeding
a
fixed
value
to be selected.
The
pari-mutuel
system
employed
at
the race-tracks
requires
explanation.
All
the
money
bet
on
prospective
winners
of a
race
is
pooled
and
a
percentage
(ranging
from
7
to
15)
known
as the 'take' is
deducted as
revenue
for the track and
for
state or
local
government.
The holders of the
winning
tickets receive the
remainder.
The
odds on
any
horse
shown on the
totalizator
at
the
track are
computed by
sub-
tracting
the
sum
bet
on
the horse
from
the
pool
after take
and
dividing
by
the sum
bet on
the
horse.
Thus
for each
dollar
the
amount
actually
returned is the
odds
plus one,
the
one
being
the dollar
bet.
To
preserve
the
relationship
of
the
mathe-
matical
expectation
as
being
the
amount
wagered
divided
by
the
odds,
where the
odds
are
the
reciprocal
of
the
probability,
this value
of the
track-odds-plus-one
will
be
referred
to as the
odds
throughout
this
paper.
When
betting
is
begun
on
a
race,
experts'
opinions
of
the
true
odds on
each horse
are
shown
and
the
totalizator
keeps
the bettors
informed,
at about
90-second
inter-
vals,
until
the
race
starts,
as
to
the
correct
odds
calculated
approximately
from
the
actual
money
bet.
No
pennies,
and in
some
states,
no
nickels are returned
on a
two-dollar
bet,
the
stray
change
going
into
another
track or
state
fund.
In
the
par-
lance
of
the
race-course this
is
termed
'breakage'
to
the
nickel or dime.8
In
this
study,
the
winners and
number
of
horses
at each
odds
were
tallied
for
519
races of
the
1947
spring
meets
of
Churchill
Downs,
Belmont
and
Hialeah
and
for
selected
odds-groups
(one
to
eleven
and
sixteen
to
twenty-one)
for an
additional
867
races
of
all
tracks
for the
first
eight
days
of
August
1947.9
All data
were
proportionately
weighted
8
For
a
more
complete
description
of the
pari-mutuel system
see
Steigleman,
op.
cit.,
17-27.
9Daily
Racing
Form
Chart
Book,
53,
1947,
nos.
1, 3, 4,
and 11.
291
as
to have
arisen from
1386
races,
that
for
odds
groups
11-16
and
above
21
being
multiplied
by
the ratio
1386/519.
If
the
psychological
odds
equaled
the a
posteriori
given by
the
reciprocal
of the
percentage
winners,
the
product
of
the
number of
winners
and
their
odds would
equal
the
number
of entries
at
each
odds-group
after
correction of the
odds had
been
made for
loss
due
to
breakage
and
take.
If
the
product
exceeds
the
number of
entries,
the
psychological
odds
were
too
large:
if the
product
is less
than
the number
of
entries,
the odds
were
too
small.
Breakage,
causing
a
larger
percentage
loss
at the
small
odds
than
at
the
large,
was
adjusted-for
by
assuming
continuous
odds
rather
than
odds
falling
into
five-cent
intervals. The
average
take,
proportionate
at all
the
odds,
was
computed
from the
data
as
being
13%.10
The
arithmetic
mean
of
the
geometric
means of
the
odds
posted
at
the
track
for each of a
sample
of
42
races
was
obtained
as
10.5;
the theoretical
minimum
for
the
geometric
mean of the
track
odds
in a
single
race
with
nine horses-the
average-size
field
in this
study-is
approximately
7.8,
and
both
values
are,
of
course,
larger
if
track-odds-plus-one
are used.
Results.
The
results
are
best shown
by
Fig.
1.
Here
it
may
be seen that
the
curve formed
by
the
product
of
the number
of winners
times the
actual
odds falls below
the
curve for
the
number of
entries
throughout
the
series.
When,
however,
the
odds
are corrected
for
breakage
and
take,
the
product
consistently
exceeds
the number of
entries
at
the
short
odds
and is
less
than the
entries
at
the
longer
odds.
The
point
of
equality
by
linear
interpolation
is
6.1,
or
at a
probability
value
of
0.16.
This
indif-
ference-point
in
track
odds
(to
which the bettors
react)
is
5.1,
with
a
reciprocal
of
0.20.
Thus below 0.16 or
0.20
the
translated
psychological
probabilities
are
too
large;
above,
too small. The
reciprocal
of
the
average
geometric
mean
of
the
track odds
of
a
sample
of
races is
0.095,
and
the
maximum for
any
one nine-horse
race is
0.13.
Discussion. All
the
factors,
rational and
irrational,
which
enter
into
the
determination of the
odds
on horses tend to
operate
in
such
a
manner
as
properly
to measure the horses' chances.
Errors
in
separate
events
cancel
in the
average
so
that
the
psychological
odds
almost
coincide
with
the
a
posteriori
odds.
The
near
congruence
of the curves
of
Fig.
1
attests
to the
skill of the
handicappers
and
patrons
at
handicapping
the horses.
10
In
any
race it
may
be shown
that the
relationship
exists between
the
odds
and
the
take
of
/
1/0
0
+
1/03
...
.+
I/On
=
1/(1
-
y)
where
0i, 02,
03,
.
.. On
are the
track-odds-plus-one
for n
horses
corrected
for
breakage,
and
y
is
the fractional
take.
The
average
value for
y
for
several
races
follows
from
summing
both
sides
of
the
equation.
292
GRIFFITH
ODDS
ADJUSTMENTS
BY BETTORS
Just
as
necessary
and
psychologically
as
remarkable,
the
money
wagered
at
each
odds
is
such as
to
preserve
the
relationship.
The final
odds are
the
summative
result of both the skill in
estimating
chances,
i.e.
adjust-
ing
the
odds,
and
also
the
proper
proportion
of the
patrons
wagering
the correct amounts
at
each odds to
preserve
them. That the number
of
bettors
preferring
each
odds-group
and the amount
they
bet are in
the
ooo0-
---o---
WINNERS
X
ODDS
CORRECTED FOR
LOSS
TO
TRACK
-----
TOTAL ENTRIES
---
0----WINNERS
X
ODDS
WINNERS
0
ODDS
FIG. 1.
NUMBER
OF
ENTRIES,
WINNERS,
AND
WINNERS-TIMES-ODDS
FOR EVERY
ODDS-GROUP
correct
proportion
may
be the
result of
several
ideologies
or
to
one
group
with the
conscious
system
of
betting
on
horses on
which
the track odds
become more
than
the
handicappers'.
By
the nature
of the
odds,
a
greater
amount is
wagered
on
the
short-
odded
horses than the
long-odded;
though
the amount
is
greater,
it
is
not
great
enough,
and
too
much
is
wagered
on
long-odded
horses.
A
direct
comparison
with
the
results of
Preston and
Baratta
is
made difficult
by
the
complexity
of
the situation. At the
same
time,
whether the
effect
is
regarded
as
a
constant error in the
judgment
of odds
or
as
a
preference
of
odds,
the
effect
may
be considered
the
same as
Preston and
Baratta's,
with even the
point
of indifference
being
approximately
the
same.
That
the
principle
holds
in
this
non-laboratory
situation
suggests
its
universality.
293
The
fundamental
difference
of
dealing
with
odds
instead
of
probabilities
is
just
one
feature
which
made
the
outcome less
predictable.
That
the
point
of
equality
should be
approximately equal
to
that
of these
writers
(cf.
footnote
p.
292)
when
it is
translated into
probability-values
calls
for
further consideration. The
nearness
of
this
value
to
theirs,
whereas
any
measure of
the
geometric
mean
is
widely
divergent, suggests
a
constancy
of
the
point
of
indifference which
may
be
relatively
independent
of the
geometric
mean.1l
SUMMARY
AND CONCLUSIONS
(1)
The
socially
determined odds on
horses
in
races
are,
on
the
average,
correct reflections of
the
horses' chances.
(2)
There
is, however,
a
systematic
undervaluation of
the
chances
of
short-odded
horses and overvaluation
of
those of
long-odded
horses.
(3)
The
indifference-point
occurs
at odds
of 6.1
or
of 5.1
when
track
odds are
used.
(4)
When these
values for the
indifference-point
are
reciprocally
trans-
lated
into
probabilities,
they
correspond
to
that
found
in
a
recent
experi-
mental
work as
between
0.05
and
0.25.
(5)
Contrary
to
the
study
cited
as
relating
this
principle
to
adaptation-
level
theory,
it
is
suggested
that this
indifference-point
may
not be
related
simply
to
the
geometric
mean
of the
choices.
"
Since
this
manuscript
was
submitted
the
statistical work
was
repeated
upon
all
the
races of
August
1934.
While our
primary
aim
was not to corroborate
the results
of
the
original investigation,
that
end
was also served.
The
results
were
nearly
as
identical
as
could
have
been
expected
had
they
arisen
from
the
same
data;
the
curves
were
practically
congruent
and the
indifference
point
fell
at
0.18
compared
to 0.16.
If
the
theory
of
adaptation-level applies
to
this
error
in
judgment,
the amount
of
money
the bettors have
(Preston
and
Baratta's
"endowment
background")
should in-
fluence the
indifference-point.
It
was
assumed
that
in
1934
the
bettors
were
in
a
state
of
impecunity
relative to their lush
affluency
of
1947
and
the
study
was
made
to
de-
termine
any
difference
in
their
adjustment
of odds. That
the
point
proved
invariant
under
these diverse
economic
conditions
lends
weight
to the
suggestion
in the
original
study
of
an
unexpected
constancy
of the
point
of
indifference.
294
GRIFFITH
The "objective probability" is determined empirically. Since the true probability of a specific horse winning is unknowable in advance, the author approximates it a posteriori (after the fact) by grouping horses with similar betting odds and calculating the percentage that actually won.
This contrasts with the betting odds, which represent the subjective or psychological probability assigned by the bettors before the race.
In 1937, Zenith Radio Corporation sponsored a nationwide radio experiment to test telepathy. "Senders" in Chicago concentrated on randomly selected Zener cards (bearing symbols like circles, stars, and wavy lines), while thousands radio listeners attempted to "receive" these thoughts and mail in their guesses.
Goodfellow analyzed over one million responses and found that above-chance "hits" were not evidence of ESP (Extra-Sensory Perception). Instead, people have systematic preferences when guessing randomly - 35% chose circles first, 78% chose heads over tails, and most favored "light" over "dark”. The apparent telepathy simply reflected moments when the random selection happened to match these common biases.
Griffith cites this to illustrate how subjective beliefs about probability - not actual probabilities - drive behavior even in controlled experiments.
The late 1940s marked the peak of horse racing's golden age in America. By the 1950s, horse racing was one of the most attended spectator sports—drawing nearly 20 million fans annually. Horses like Seabiscuit, Citation, and Whirlaway were national celebrities, and Triple Crown races received front-page newspaper coverage.
This was also the only legal gambling game in town. Casinos were confined to Nevada, state lotteries didn't exist, and sports betting was illegal everywhere. Horse racing had a virtual monopoly on legal wagering, making tracks both entertainment venues and de facto casinos.
Before betting opened, track handicappers posted "morning line" odds—their expert predictions of each horse's chances. These served as a starting point for bettors.
The totalizator (or "tote board") was a mechanical calculating machine that continuously tracked all bets placed and computed the current odds for each horse. Invented in 1913 by Australian engineer George Julius, it was originally designed as a vote-counting machine before being adapted for racetracks. The totalizator displayed updated odds roughly every 90 seconds, allowing bettors to see how the pool was shifting as post time approached.
Pari-mutuel (French for "mutual betting") means bettors wager against each other, not against the house. All bets are pooled together, the track takes a fixed percentage, and the rest is distributed to winners.
Example: Consider a race with three horses where 10,000 USD total is wagered:
- Horse A: 5,000 USD bet
- Horse B: 3,000 USD bet
- Horse C: 2,000 USD bet
The track takes 15%, leaving 8,500 USD to pay winners.
If Horse C wins, the 8,500 USD is split among those who bet on Horse C. Each 1 dollar bet on Horse C returns $ \$8,500 ÷ \$2,000 = \$4.25$. This means odds of 3.25-to-1 (you get \$3.25 profit plus your \$1 stake back).
**Key insight:** The odds aren't set by the track - they emerge from how bettors distribute their money. If more people pile onto a horse, its odds drop; if money flows elsewhere, its odds rise. The totalizator displays these constantly updating odds so bettors can see the current state of the pool.
**Breakage** refers to rounding payouts down to the nearest nickel or dime. On a \$2 bet, a calculated return of \$7.37 might pay only \$7.30 - the track keeps the difference.
**The Theoretical Minimum (7.8)**
The value **7.8** serves as a mathematical baseline for completely unpredictable race - a hypothetical scenario where the crowd believes every single horse has an exactly equal chance of winning.
**The Calculation**
The author derives 7.8 by assuming the betting pool is split perfectly evenly among the average field size of 9 horses, adjusted for the track's revenue deduction ("take" - 13%):
$$
\text{Odds} = \text{Field Size} \times (1 - \text{Take})
$$
$$
\text{Odds} = 9 \times (1 - 0.13) = \mathbf{7.83}
$$
**Why it Matters**
The fact that the actual observed average (**10.5**) is significantly higher than **7.8** proves that bettors are not guessing randomly. By heavily backing favorites (low odds) and ignoring longshots (high odds), the crowd creates variance in the market, which mathematically forces the geometric mean up from its theoretical minimum.
In standard probability theory, odds are defined as the reciprocal of probability: if an outcome has probability p, its fair odds are $\frac{1}{p}$, and the expected value of a wager is the stake multiplied by $p$ and divided by the odds. In pari-mutuel betting, however, the odds displayed at the track usually omit the return of the original stake, showing only the net profit. By redefining “odds” as track-odds plus one, the paper restores the conventional mathematical relationship between odds, probability, and expectation. With this definition, an even bet corresponds to odds of 2, matching a probability of $\frac{1}{2}$, and expected value calculations remain consistent throughout the analysis.
Griffith's test works as follows: if bettors correctly assess probabilities, then for any odds group:
$$\text{Number of winners} \times \text{Odds} = \text{Number of entries}$$
Across hundreds of races, Griffith recorded each horse's odds and whether it won. He then grouped all horses by their odds. If 100 horses across different races all went off at odds of 5 (implied probability 20%), we'd expect about 20 of them to have won.
**Observed results:** The product (winners × odds) fell below entries at every odds level - simply reflecting the track's take.
After adjusting for take and breakage, a pattern emerged:
- **At short odds (favorites):** winners × odds > entries → favorites won *more* often than bettors expected
- **At long odds (longshots):** winners × odds < entries → longshots won *less* often than bettors expected
The **indifference point** - where subjective and objective probabilities align - occurred at odds of 6.1 (probability ≈ 0.16).
This is the favorite-longshot bias: bettors systematically underbet favorites and overbet longshots. A dollar wagered on favorites yielded better returns than a dollar wagered on longshots, even though both lost money on average due to the take.
Elizabeth Becknell's 1940 paper argued that human decisions are often driven by "ideologies" or internal concepts of chance rather than actual statistical likelihoods.
Two individuals facing identical odds may act very differently based on their beliefs about chance and uncertainty.
The term "ideologies of probability" refers to the mental frameworks people use to interpret uncertain events - some may believe in luck or hot streaks, others in pure mathematical randomness. These belief systems shape probability judgments in ways that deviate from objective calculation.
Preston and Baratta (1948) ran laboratory experiments where subjects bid on uncertain gambles. They discovered a systematic pattern in how people perceive probabilities:
- Small probabilities are overvalued (a 5% chance "feels" larger than 5%)
- Large probabilities are undervalued (a 90% chance "feels" smaller than 90%)
The indifference point - where subjective and objective probability match- fell near the geometric mean of the probabilities they tested (around 0.20-0.25).
They linked this to Helson's adaptation-level theory, which proposes that humans judge stimuli not in absolute terms but relative to a neutral reference point. This reference point is determined by the range of stimuli encountered, and Helson found it typically falls at the geometric mean of that range.
Horse-race betting typically follows a pari-mutuel system. The term comes from the French pari mutuel, meaning “mutual bet” or “bet among ourselves,” emphasizing that bettors wager against one another rather than against the house. All bets on a race are placed into a common pool, from which the track and the state remove a fixed percentage as revenue. The remaining money is then shared among those who bet on the winning horse.
Because payouts come from this shared pool, the odds on each horse are determined entirely by how the total money is distributed. For example, if \$10,000 is wagered in total and \$1,000 is placed on a particular horse, that horse represents roughly one tenth of the pool and will have relatively high odds. If additional bettors move their money to that horse, its share of the pool increases and its odds fall. The odds therefore summarize the crowd’s aggregate judgment rather than an externally assigned probability, which is why they are described as socially determined.