vpFREE2 Forums

"E.R."

#1

At the risk of beating a dead horse, I pose the following problem --
since concrete examples sometimes speak more strongly than general
discussion ...

···

----------

Over a period of time, 10000 players approach a $1 game that has an
expected return (ER) of 101% when played with max-ER strategy.
They're each staked with $2500. They play 10,000 hands and return the
proceeds. If they bust, they report the hand at which they bust.
(And, assume 100% integrity here, ok? :wink:

Statistically it's expected that 1000 of the players will bust and
that they will do so at their 8000th hand on average.

----------

Of the players who bust, what is their expected return from their play
(coin out/coin in)? (note: obviously, coin in - coin out = 2500)

Of the players who play the full 10,000 hands, what is their expected
return from the play?

From the casino's perspective, what is their expected hold under these
assumptions (express as coin held/coin in)?

- Harry

#2

Since the return is 1.01 then if x is the average amount for the
nonruin people the

1.01 = (-1000(2500)+9000x)/(10,000+8000)

so x = (1.01(18000) +1000(2500))/9000 =
279.798
and their ER = 1 + 279.798/10000 = 1.0279798

where do I pick up my prise!

DB

-- In vpFREE@yahoogroups.com, "Harry Porter" <harry.porter@v...>
wrote:

···

At the risk of beating a dead horse, I pose the following problem -

-

since concrete examples sometimes speak more strongly than general
discussion ...

----------

Over a period of time, 10000 players approach a $1 game that has an
expected return (ER) of 101% when played with max-ER strategy.
They're each staked with $2500. They play 10,000 hands and return

the

proceeds. If they bust, they report the hand at which they bust.
(And, assume 100% integrity here, ok? :wink:

Statistically it's expected that 1000 of the players will bust and
that they will do so at their 8000th hand on average.

----------

Of the players who bust, what is their expected return from their

play

(coin out/coin in)? (note: obviously, coin in - coin out = 2500)

Of the players who play the full 10,000 hands, what is their

expected

return from the play?

From the casino's perspective, what is their expected hold under

these

assumptions (express as coin held/coin in)?

- Harry

#3

DB wrote:

Since the return is 1.01 then if x is the average amount for the
nonruin people the

1.01 = (-1000(2500)+9000x)/(10,000+8000)

so x = (1.01(18000) +1000(2500))/9000 = 279.798
and their ER = 1 + 279.798/10000 = 1.0279798

where do I pick up my prise!

DB

Much appreciated, DB.

There's a modest slip in that you reference hands played rather than
convert to coin-in for the ER calculation.

Just to spell out in "long hand":

> ----------
>
> Over a period of time, 10000 players approach a $1 game that has
> an expected return (ER) of 101% when played with max-ER strategy.
> They're each staked with $2500. They play 10,000 hands and return
> the proceeds. If they bust, they report the hand at which they
> bust.
>
> Statistically it's expected that 1000 of the players will bust and
> that they will do so at their 8000th hand on average.
>
> ----------
>
> Of the players who bust, what is their expected return from their
> play (coin out/coin in)?

Expressed on a per player basis:
Coins in on average = 8000 * 5 = 40000
Coins out = 40000 - 2500 loss = 37500
E.R. of those who bust = 37500/40000 = 93.75%

(and, as a reminder, this is purely a reference to an expected result.
It implies know foreknowledge of how those who ruin will perform, and
how many will actually ruin.)

> Of the players who play the full 10,000 hands, what is their
> expected return from the play?

We know that of the expected 9000 who complete their play, a total of
90 mil. hands will have been played. That's total coin in of $450 mil.

For the group of 1000 who are expected to bust, on average of 8000
hands are expected to be played. That's 8 mil. hands and coin in of
$40 mil.

Total expected coin in of the entire field of 10000 players is $450 +
$40 = $490 mil.

As a whole, the results of the players will be expected to conform to
the stated 101% ER of the game. Expected coin out will be $490 mil. *
1.01 = $494.9 mil, for a net gain of $4.9 mil.

Because we expect that the subset of players who bust will have a
loss of $2500 each, or $2.5 mil for the 1000 players in total, it's
expected that the 9000 players who play through the full 10,000 hand
trial will profit by $7.4 mil. (7.4 win - 2.5 loss = 4.9 net win)

Therefore, this subset, with expected coin in of $450.0 mil and coin
out of $457.4 mil, will have an ER of 457.4/450.0 = 101.64%.

> From the casino's perspective, what is their expected hold under
> these assumptions (express as coin held/coin in)?

This is the crux of the exercise: To show that when an ER for each of
the two player subsets is expressed as above, those values can be
reconciled to the ER of the casino.

The casino ER is a no-brainer: it's 99%. ($99 won for every $100
wagered ... for the players it's $101 won for every $100 wagered).

The reconciliation of players' ERs is (using an weighted average of
the above ER's, total hands - 98 mil - played by each group being
weighted):

93.75%*(8 mil/98 mil) + 101.64%*(90 mil/98 mil)
7.653% + 93.343% = 101.0% (cut be a break on the .004% rounding
difference, ok? :wink:

- Harry

#4

Harry Porter wrote:

snip

Of the players who bust, what is their expected return from their play
(coin out/coin in)? (note: obviously, coin in - coin out = 2500)

Their expected return _WAS_ 101% while they were playing. Since the games have been played and their bankroll exhausted, they no longer have any _expectation_ regarding that particular money. Their ACTUAL return was zero.

Of the players who play the full 10,000 hands, what is their expected
return from the play?

Same as above; while they were playing, their expected return _WAS_ 101%, but once they stop they no longer have any expectation, and their ACTUAL return was whatever it was.

Now, Harry, in the interest of hopefully settling this once and for all, will you pay me the courtesy of allowing me to give you a different hypothetical. Two brothers have been fishing together everyday for the past 30 years, and have been diligently keeping statistical records; 'Harry' has averaged 5 fish per day, and 'Porter' has averaged 6 fish per day. Today they went fishing, and when they came back to the dock a friend asked them about their catch. Harry said that his EC (expected catch -- what he 'expected' to catch based on statistics) was 5 ... BUT ... his AC (actual catch) was only 3 fish; Porter, on the other hand, had better luck, and he explained that his EC was 6, because he 'expected' to catch 6 fish which is his daily average, but he happened to hit the jackpot and so his AC (actual catch) was 9. 'Expectations' and 'Actuality' are seldom the same.

I _promise_ that this will be my last post on this subject.

Cheers.

Bill Velek

#5

Bill Velek wrote in reply to:

> Of the players who bust, what is their expected return from their
> play (coin out/coin in)? (note: obviously, coin in - coin out =
> 2500)

Their expected return _WAS_ 101% while they were playing. Since the
games have been played and their bankroll exhausted, they no longer
have any _expectation_ regarding that particular money. Their
ACTUAL return was zero.

Two points:

It's important to understand that the problem is stated at the outset
of the trials. There are no actual results yet. But clearly it's
anticipated that the players will ultimately divide into two groups:
those who bust and those who survive the trial with their bankroll intact.

Statistically a statement of the expected return of these two discrete
groups can be made in advance of play. That expected return in each
case will differ from that of the entire body of the players as a
whole (which will equal the ER of a game when no other information is
provided).

Second: The actual return of any player who busts isn't zero. Return
is coin-out/coin-in. This value is equal to 0 only when each and
every play fails to produce a win.

Failure to agree on these points will preclude any additional discussion.

- Harry

#6

And this is the conundrum as I see it. Of course every vp player understands the difference between expectation and actuality --Sometimes in the most painful way. If I'm understanding what I'm not understanding is how you reconcile a finite vp reality (whether it be a win or loss, expected or unexpected) with a mathematical ideal based on an infinite number of hands. Can what has happened and what is likely to happen be associated in a meaningful way? I have wavered between disagreement and confusion and will likely continue to do so;-)

Chandler

···

At 10:37 PM 7/25/2004, you wrote:

'Expectations'
and 'Actuality' are seldom the same.

#7

Chandler wrote:

>'Expectations' and 'Actuality' are seldom the same.

And this is the conundrum as I see it. Of course every vp player
understands the difference between expectation and actuality --Sometimes in
the most painful way. If I'm understanding what I'm not understanding is
how you reconcile a finite vp reality (whether it be a win or loss,
expected or unexpected) with a mathematical ideal based on an infinite
number of hands. Can what has happened and what is likely to happen be
associated in a meaningful way? I have wavered between disagreement and
confusion and will likely continue to do so;-)

Since I wasn't planning to continue this discussion, I refrained from answering your questions until other folks had plenty of time to do so; now I'll answer because I think your question deserves an answer from someone.

"Can what has happened and what is likely to happen be associated in a meaningful way?" The short answer is 'No'; nothing that has already happened, in truly random video poker, can ever change the probability of what is going to happen in the future. Period. I don't care if you were lucky enough to have just hit a dozen Royals in a row; the probability that you'll get a Royal on the next hand is exactly the same as if you have not seen a Royal in the past million hands.

This is precisely why I find the recent discussion with Harry Porter so frustrating; it's because I realize that he knows that fact very well, and I know he accepts it without any question, so I just don't understand why he is proposing that ER changes with changes in circumstances -- because even if it's just a matter of semantics (I'll give an example in a minute), utilization of such a fact is completely without usefulness or any practical value.

For instance, let's say that I have made plans to play exactly 1000 hands of vp today, and being a little off in the head, I'll stick to my plan no matter what happens. We head into a casino where I know that there are some vp machines which have an ER of 101% with perfect play, and I know the game like the back of my hand. However, as we go in, my wife 'needs to see a man about a horse' ... :wink: ... and while I'm waiting, standing there next to an absolutely horrible vp machine with an ER of only 95%, against my better judgment I play my first game ... and hit a Royal Flush. Now if I understand what Harry is saying, we can factor in the circumstances of that game as part of the limited number of 1,000 games that I'll be playing, in order to see what sort of ER (I think he means AR -- actual return) that I'm likely to have during just that session. I will concede that there is a grain of truth to what he proposes, because I can now predict, before playing the remaining 999 games, that my ACTUAL Return, in the end, will almost certainly be considerably above the mathematically correct long-term ER for that machine. Fine; perhaps that makes me feel good, but other than that there is no use or value to that information. For instance, at that point on that 'normally-95%-ER' machine, after winning a Royal on my first game, I might use Harry's method to conclude that my ER 'on that machine for 1,000 games' will be 174.9% -- but so what? Am I going to keep on playing that lousy 95% machine because my 'session ER' is 174.9%? Of course not, if I have any sense. Or am I ever going to make any strategy changes because I've re-determined my ER according to whatever circumstances? Of course not. But someone who doesn't know what they're doing, and doesn't really understand the implications of thinking that the ER changes, might do the wrong thing -- stick with a bad machine or change their strategy -- and that's why I think it's important to understand that ER is _NEVER_ changed for future events, and anything that has already happened is no longer 'expected', but rather is it 'actual'.

I hope this clarifies the matter for you.

Cheers, and good luck.

Bill Velek

···

At 10:37 PM 7/25/2004, you wrote:

#8

Bill Velek wrote:

Now if I understand what Harry is saying, we can
factor in the circumstances of that game as part of the limited
number of 1,000 games that I'll be playing, in order to see what
sort of ER (I think he means AR -- actual return) that I'm likely to
have during just that session.

<snip>

For instance, at that point on that 'normally-95%-ER' machine, after
winning a Royal on my first game, I might use Harry's method to
conclude that my ER 'on that machine for 1,000 games' will be 174.9%
-- but so what? Am I going to keep on playing that lousy 95%
machine because my 'session ER' is 174.9%?

Bill,

If this were an accurate reflection of my statements I think it would
be apparant that I've gone off my rocker.

Based upon past exposure to my contributions here and elsewhere, I
would have hoped you might question that suggestion. The alternative
would be to take another look at what I've written to see if you
haven't fully interpreted what I've written.

There's no need to continue our discussion on the topic. But I wanted
to suggest that there's something a little out of whack here.

- H.

#9

Harry Porter wrote:

Bill Velek wrote:
> Now if I understand what Harry is saying, we can
> factor in the circumstances of that game as part of the limited
> number of 1,000 games that I'll be playing, in order to see what
> sort of ER (I think he means AR -- actual return) that I'm likely to
> have during just that session.

I said "... _IF_ I _understand_ ..." what you are saying, but from your reply I obviously don't. I've tried to understand, and I've attributed our lack of agreement to semantics, but I've pretty much given up now because I don't think it's important enough to warrant spending time trying to sort it out.

<snip>

> For instance, at that point on that 'normally-95%-ER' machine, after
> winning a Royal on my first game, I might use Harry's method to
> conclude that my ER 'on that machine for 1,000 games' will be 174.9%
> -- but so what? Am I going to keep on playing that lousy 95%
> machine because my 'session ER' is 174.9%?

Bill,

If this were an accurate reflection of my statements I think it would
be apparant that I've gone off my rocker.

Based upon past exposure to my contributions here and elsewhere, I
would have hoped you might question that suggestion. The alternative
would be to take another look at what I've written to see if you
haven't fully interpreted what I've written.

I'm sorry, Harry, but I _have_ looked at your posts -- carefully -- and, for the very first time that I can ever remember, I can't figure out what you are saying other than the way in which I had interpreted it. The examples you have used have not clarified your position for me. My examples have obviously failed in your direction, too -- although I thought that what fishermen 'Harry' and 'Porter' _EXPECTED_ to catch versus what they _ACTUALLY_ caught would have completely cleared that up. Other respected contributors have similarly pointed out the distinction between _expected_ return (future expectancy) and events that have actually occurred. But, in the immortal words of the warden in the movie: 'Cool Hand Luke', "What we have here is a failure to communicate."

There's no need to continue our discussion on the topic. But I wanted
to suggest that there's something a little out of whack here.

At any rate, I apologize if I portrayed you and/or your position on ER in any way that is inaccurate. I like you personally, I enjoy your posts, and appreciate your many valuable contributions, and, to say the least, I was surprised at the position that I _thought_ you had taken, and you will note that I also commented that you certainly know better. At any rate, in the final analysis, I will make one unequivocal statement: ER based upon correct long-term application of the appropriate strategy NEVER changes, and therefore the correct strategy for a given goal never changes, either. So long as we can agree on that, then I'm happy; if we don't agree, then naturally I think that you are wrong.

Cheers.

Bill Velek