vpFREE2 Forums

Please explain "equation"

#1

vpFae6128305 wrote:

On 21 Nov 2005 at 14:07, Dan Paymar replied:

Obviously I missed that campaign. My problem with it is that it flies
in the face of the standard mathematical definition of Expected
Value, which is always unit based.

For example, an EV=0.9 means that you
can expect a return of 90 percent of whatever amount you bet. Thus,
EV is independent of the size of the bet or the number of coins.

First you say that EV is unit based and now you're saying that
EV is a percentage figure.

EV is a mixed number, to use the math term. If EV=1.0000, then your expected value is equal to your wager, and in the long run you can expect to break even. If EV is less than one, it's a losing proposition. If EV>1, you can expect to win in the long run. Saying that an event has an EV equal to 0.9 is saying that, over the average of many such events, you can expect to get back 0.9 times your wagers. That's the same as saying 90% of your wagers. It's just two ways of saying the same thing. 0.9 and 90% mean exactly the same thing. That's just the definition of "percent."

Another example. A game is based on the flip of a perfectly balanced coin. Tails, you lose your bet, heads you get back twice your bet (2-for-1). I think you will agree that in the long run you can expect to break even on this game. Now suppose your bet is five coins. Is the EV for that bet five? No! The EV is still one. That's according to the mathematical definition of EV. If your long term expectation is to break even, then the EV is one no matter what is bet.

It doesn't matter whether a 9/6 JoB is a nickel machine or a dollar machine, whether it's a five-coin or ten-coin machine, or whether it's single play or 100-play; the EV is 0.99544 with perfect strategy and max coins played.

I think it was jazbo who made it clear a few years ago that the EV of a game (which Lenny Frome called "payback") is exactly equal to the EV of the next play before the cards are dealt.

I believe your example is (should be) referring to ER rather than EV.

I believe that my example was correct as stated.

If you're right, mathematically speaking (and it doesn't sound right
to me), it shouldn't be extended to video poker (and isn't on vpFREE).

Any mathematically defined term can, and should, be applied to video poker, and to any other game for which probabilities can be calculated.

See the vpFREE Expected Value (EV) Usage Poll:

<http://members.cox.net/vpfree/P_EV.htm>

Polls are too often weighted by people who know little about the subject, so I don't give any of them them much credence.

EV should be expressed in units and ER should be expressed
as a percentage figure and their relationship is explained by:

EV = [ER] X [coin-in]

I agree that EV should be expressed in units, but this formula says that EV is a number of coins. On a five-coin quarter machine, $1.25 is one unit, not five.

> If vpFREE wants to use a different definition of EV, I can't change

that, but it seems that it would confuse some readers.

IMO, what confused readers, players and a few VP gurus was the
former interchangeable usage of EV and ER . That confusion has
been eliminated on vpFREE.

How can confusion be eliminated by redefining well-defined mathematical terminology?

A quick search with Google turned up millions of references to Expected Value. I scanned several of them, and none says anything about the amount of a wager. EV is always relative to the amount bet.

Expected Return, however, shows definitions such as "An estimation of the value of an investment." Thus if five coins are "invested" in one play on a video poker machine with an EV of 1.01 (i.e., 101% "payback"), then you can expect a long term average of 5.05 coins back per play. If you want to call that ER, that's OK, but that would make your equation

ER = EV x coin-in

To best understand each other, it's obviously necessary to agree on terminology. It's great to set standards, but those standards should not conflict with definitions already long established with wide acceptance in the scientific community.

Please take this as a suggestion, not as criticism.

Best regards,
Dan

···

--
Dan Paymar
Author of best selling book, "Video Poker - Optimum Play"
Editor/Publisher of VP newsletter "Video Poker Times"
Developer of VP analysis/trainer software "Optimum Video Poker"
Visit my web site at www.OptimumPlay.com

"Chance favors the prepared mind." -- Louis Pasteur

#2

The EV = [ER] X [coin-in] equation is correct as stated
in the FAQ and as I explained it, with EV being expressed
in dollars and ER being a percentage figure.

However,

Dan Paymar (mathematical terminology?) and vpFREE have
reversed the definitions of EV and ER.

Dan Paymar (mathematical terminology?) says:

... How can confusion be eliminated by redefining well-defined
mathematical terminology? ...

... For example, an EV=0.9 means that you can expect a
return of 90 percent of whatever amount you bet ...

... Expected Return, however, shows definitions such as
"An estimation of the value of an investment ..."

This says that Expected VALUE is a RETURN expressed in
percent, and that Expected RETURN is a VALUE.

This seems backwards.

vpFREE's terminology makes more sense to me.

However, I assume that Dan's mathematical definitions are
correct, but will appreciate verification from others. Then perhaps
we should consider conforming, or not. I note that jbqueru (who
seems to be a math person) used EV and ER in the same
context as vpFREE.

vpFae

#3

I was trying really hard to stay out of this, but since you asked
for "verification from others..."

The mathematical definition of "expected value" is inherently
ambiguous. There is no _requirement_ that it be expressed as
either a pure number or as a percentage. Mathematically, the
"expected value" of a random variable takes on whatever units
are associated with what is being measured, and this can be
just about anything imaginable, as long as it can be quantified
in a meaningful way.

Example: X can be a random variable that represents height,
in inches, of the "average" person in the United States. This
has units of inches, but one could do an equivalent computation
in feet, centimeters, furlongs or angstroms and it wouldn't really
change the result. The definition of "expected value" of any
random variable is the same thing that we call the mean. The
units are either a natural result of the problem, or a _choice_
made by whoever is framing the problem.

Most applications of probability have absolutely nothing to do
with gambling or wagers. So, claiming that EV and or ER are
"well defined mathematical terminology" is somewhat specious.
At best, there may be defacto standards as to what these terms
mean, but if such standards are perceived to exist, they are
the invention of gambling "experts" rather than mathematicians.

In addition, there is nothing in the math that requires the original
bet to be factored into the computation in any way. If a casino
were to offer a "free pull" to people walking past (say, at the
Four Queens in Vegas, or at any of a number of other casinos)
then there isn't even a wager to take into consideration. Yet,
we can still computed the expected value of a variable which
represents the number of dollars (or quarters, or Big Macs) that
fall out of the machine after a favorable outcome.

Bottom line, it seems naturable that EV mean "expected value"
and this begs the question as to what random variable X we are
using to compute E(X). Expected return seems to naturally suggest
that we are computing the amount of money returned by the machine
whenever it decides not to keep the entire wager. This is different
than performing a computation of expected GAIN (which means net
win/loss) but the two are mathematically equivalent. All of these
things can be expressed as a percentage of the wager, but that is
a convention that some find convenient, it is not strictly required
by any "official" definition (except one that might be chosen, say,
for a particular forum...).

VP is nice in one sense -- the original wager is almost always the
entire wager, so there is no ambiguity in the term "wager." Other
games are less clear. In blackjack, the player can split and/or
double down, by placing additional wagers after some cards have
been revealed. Craps allows odds bets to be added/removed at
any time. For such games, the expected outcome can be computed
by ignoring the wagers entirely and computing the net gain as it
is perceived at the end of one round of play. This is the method
I use for doing combinatorial analysis of blackjack.

Ultimately, I think we're stuck with terminology that isn't universal
and probably never will be. In order for clear communication to
take place, it is best for authors to either define their terms or
at least give a reference to a commonly available guideline for
whatever terminology he/she will use. But even when authors
don't bother to do this, it is usually possible to figure out from
the context of the discussion what is meant by their use of
the terms EV and/or ER, as long as the terms are used in a
reasonably consistent fashion.

···

On Monday 21 November 2005 10:30 pm, vpFae6128305 wrote:

The EV = [ER] X [coin-in] equation is correct as stated
in the FAQ and as I explained it, with EV being expressed
in dollars and ER being a percentage figure.

However,

Dan Paymar (mathematical terminology?) and vpFREE have
reversed the definitions of EV and ER.

Dan Paymar (mathematical terminology?) says:
> ... How can confusion be eliminated by redefining well-defined
> mathematical terminology? ...
>
> ... For example, an EV=0.9 means that you can expect a
> return of 90 percent of whatever amount you bet ...
>
> ... Expected Return, however, shows definitions such as
> "An estimation of the value of an investment ..."

This says that Expected VALUE is a RETURN expressed in
percent, and that Expected RETURN is a VALUE.

This seems backwards.

vpFREE's terminology makes more sense to me.

However, I assume that Dan's mathematical definitions are
correct, but will appreciate verification from others. Then perhaps
we should consider conforming, or not. I note that jbqueru (who
seems to be a math person) used EV and ER in the same
context as vpFREE.

#4

I note that jbqueru (who
seems to be a math person) used EV and ER in the same
context as vpFREE.

Let's say I was a math person (I did 4 years of math and physics in
college to have a solid base before turning toward computer science
then finally software engineering). Unfortunately I studied in France,
which means that I am not familiar with the English names of most
concepts. Furthermore I've only reluctantly studied statistics and
probabilities (I've always prefered algebra, calculus and the branch
that we call arithmetic in France), so you can consider me a
near-total layman when it comes to using those names.

Now, as a layman, I agree that I feel more comfortable measuring a
"return" in percentage and a "value" in dollars. I can totally
understand that different words may have been coined by mathematicians
for those concepts in a different context, but in the context of VP I
believe that the traditional words are somewhat confusing and I
certainly feel more comfortable using the vpFREE meaning for those
concepts.

That being said, there are quite a few concepts in VP that are close
to the concepts of EV and ER with some added subtlety, and when
there's any doubt it's better to use explicit "units" (percentage or
dollars) and to use qualifiers. We could go overboard and try to
invent names for all those concepts, but that would probably be
pointless and confusing. I mean, we don't need a formal name for the
expected hourly profit excluding comps of a game, do we?

JBQ

#5

my $.02

seems to me return implies a ratio, for example for fpdw
er=1.007619612 = ratio of coin-out to coin-in

value implies a fixed value, the obvious one would be dollars, for
example the ev of a 20 coin for a quad coupon in 5 coin jacks or
better is 20 + 5 x 423.2722381 x (er-1) = 10.34736909 coins

$ev/hour = bets/hour x dollars/bet x (er-1)

···

--- In vpFREE@yahoogroups.com, "vpFae6128305" <vpfae6128305@C...> wrote:

The EV = [ER] X [coin-in] equation is correct as stated
in the FAQ and as I explained it, with EV being expressed
in dollars and ER being a percentage figure.

However,

Dan Paymar (mathematical terminology?) and vpFREE have
reversed the definitions of EV and ER.

Dan Paymar (mathematical terminology?) says:

> ... How can confusion be eliminated by redefining well-defined
> mathematical terminology? ...

> ... For example, an EV=0.9 means that you can expect a
> return of 90 percent of whatever amount you bet ...

> ... Expected Return, however, shows definitions such as
> "An estimation of the value of an investment ..."

This says that Expected VALUE is a RETURN expressed in
percent, and that Expected RETURN is a VALUE.

This seems backwards.

vpFREE's terminology makes more sense to me.

However, I assume that Dan's mathematical definitions are
correct, but will appreciate verification from others. Then perhaps
we should consider conforming, or not. I note that jbqueru (who
seems to be a math person) used EV and ER in the same
context as vpFREE.

vpFae

#6

jbqueru wrote:

I mean, we don't need a formal name for the
expected hourly profit excluding comps of a game,
do we?

bjaygold wrote:

We already have one, though I wouldn't consider it
"formal." It's called the "earn" of the game.

Another name would be "edge", and the equation would
be: Expected Profit (or loss) = Edge X Action.

After considering the public and private responses to
the appropriateness of the terminology in the vpFREE
Video Poker Equation, the Administrator and I have
decided against making any changes and will continue
with: EV = [ER] X [coin-in]

See FAQ #8 for additional details.

Thanks for the responses.

vpFae