Standard Deviation of multiple line play---Help please

[deuceswild1000]

Help from you more mathematically inclined would be appreciated. I
have read Jazbo and Wizard of Odds, but cannot completely put in to
numbers what I am trying to learn. I am hoping that someone could
take the time to help me by using more/simpler examples or different
words to get a handle on this.

To help you find the references, here are the URLs

http://www.jazbo.com/ on the left side click N-play Bank Roll

http://wizardofodds.com/videopoker/vidpokapx3.html

I want to compare expected variation of playing five line multiple
play versus single line where the total amount per push of the deal
button is equal. I will use as examples single line at $0.25/credit
= $1.25 per push of the deal button compared to five line
$0.05/credit also equals $1.25 per push of the deal button when each
are played at full coin per line. I further realize that the
concepts could be applied to higher dollar values or different
number of lines.

Not sure how to do different number of lines (5 line is tabulated by
Wizard of Odds) as that is part of the problem I am having in
understanding the two references above. Explanations here would
also be appreciated

I will use FPDW as the example as both Jazbo and Wizard of Odds
covers it in there pages referenced above. However, I would prefer
that the analysis be done on NSUD as that is available to me, but
the reference data is not available to me. Again, if someone could
also teach me how to do NSUD, that would also be appreciated.

This is what I think I know so far:

If I play an infinite number of sessions of 4900 hands/session of
one line FPDW, then I think I understand that I can expect each
session to produce on average a win (over the long term) of 0.00765
betting units (100.765 expected return) = .00765 *$1.25 * 4900 =
$46.86 per session

The standard deviation of the distribution of infinite sessions of
4900 would be sqrt (4900 * 25.835) = 355.8 betting units * $1.25 per
betting unit = $444.74 std dev in dollars.

Then assuming normal distribution (and I know that this is not
completely agreed upon), 67% of all sessions would fall within
plus/minus one std dev; 95% within +/-two std dev; and 99+% would
fall within +/- three std dev from the expected results of $46.86.

Am I correct so far?

Ok, now what if I played $1.25 per push of the deal button using
$0.05 per credit on five lines at full coin?

I assume that infinite sessions of 4900 deals of five line would
also have a distribution with mean or expected results of $46.86
per long term session and a std dev of ???

What would be the std dev of sessions of 4900 pushes of five line?
Just to make sure that I am getting my point across, this would be a
total of 4900 pushes * 5 lines/push = 24,500 hands of vp, However
I would have put the same amount of money at risk as in the single
line example of above.

Swings in session outcome putting the same amount of money at risk
($6125 each session in this example) is what I am trying to
evaluate. I realize that multiple line will probably play slower
than single line, but so be it.

If you prefer to e-mail me privately, that would be fine, but I
think others would be interested also, and some are probably just as
math challenged as me.

Please point out any wrong computations or assumptions.

DWK

[nightoftheiguana2000]

wrote:

If I play an infinite number of sessions of 4900 hands/session of
one line FPDW, then I think I understand that I can expect each
session to produce on average a win (over the long term) of 0.00765
betting units (100.765 expected return) = .00765 *$1.25 * 4900 =
$46.86 per session

The standard deviation of the distribution of infinite sessions of
4900 would be sqrt (4900 * 25.835) = 355.8 betting units * $1.25 per
betting unit = $444.74 std dev in dollars.

Then assuming normal distribution (and I know that this is not
completely agreed upon), 67% of all sessions would fall within
plus/minus one std dev; 95% within +/-two std dev; and 99+% would
fall within +/- three std dev from the expected results of $46.86.

Infinite sessions of 4900 would be a normal distribution.

Am I correct so far?

Yes.

Ok, now what if I played $1.25 per push of the deal button using
$0.05 per credit on five lines at full coin?

I assume that infinite sessions of 4900 deals of five line would
also have a distribution with mean or expected results of $46.86
per long term session and a std dev of ???

What would be the std dev of sessions of 4900 pushes of five line?
Just to make sure that I am getting my point across, this would be a
total of 4900 pushes * 5 lines/push = 24,500 hands of vp, However
I would have put the same amount of money at risk as in the single
line example of above.

Swings in session outcome putting the same amount of money at risk
($6125 each session in this example) is what I am trying to
evaluate. I realize that multiple line will probably play slower
than single line, but so be it.

If you prefer to e-mail me privately, that would be fine, but I
think others would be interested also, and some are probably just as
math challenged as me.

Please point out any wrong computations or assumptions.

DWK

Variance is a function of number of lines, for deuces:
Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line Var=~8

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

Variance is a function of number of lines, for deuces:
Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line Var=~8

So ARE YOU SAYING that the variance for five line play is
significantly smaller than for single line? Approx. 8/26ths, even
though the same amount of money is put at risk.

I envision an infinite distribution of outcomes of 4900*5 = 24500
hand sessions where the variance or std dev would would then by
multiplied by $0.25 per betting unit and in trying to follow Jazbo
and WoO I get that the std dev in dollars would be much higher than
single line at the same amount of dollars in.

Would you please post where your calculations can be learned or
found. This seems to be different than what Jazbo and WoO have
published, which so far are my only source of information.

BTW, thanks for being the only one to answer this so far. I would
like to hear from others.

DWK

[deuceswild1000]

Also would anyone have the value of covariance for NSUD?

Even further could the formula for calculating the covariance of any
game be posted?

DWK

[nightoftheiguana2000]

wrote:

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
> Variance is a function of number of lines, for deuces:
> Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line Var=~8

So ARE YOU SAYING that the variance for five line play is
significantly smaller than for single line? Approx. 8/26ths, even
though the same amount of money is put at risk.

Yes.

I envision an infinite distribution of outcomes of 4900*5 = 24500
hand sessions where the variance or std dev would would then by
multiplied by $0.25 per betting unit and in trying to follow Jazbo
and WoO I get that the std dev in dollars would be much higher than
single line at the same amount of dollars in.

Would you please post where your calculations can be learned or
found. This seems to be different than what Jazbo and WoO have
published, which so far are my only source of information.

By Variance I'm refering to Variance per bet, which makes it easier to
compare machines and is the same as the number called Variance in
programs such was Winpoker and Frugal VP. Jazbo and Wizard are
refering to dollar Variance and comparing the dollar Variance of say
single line quarter to 5-play quarter. Of course 5-play quarter has a
higher dollar variance than single line quarter just as dollar deuces
has a higher dollar variance than quarter deuces. If you make the
adjustments you will see that the numbers are actually the same.

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

BTW, thanks for being the only one to answer this so far. I would
like to hear from others.

DWK

[nightoftheiguana2000]

It's about the same as fpdw.
Yes, you can calculate the dealt variance of any game, assuming you
know the probability of dealt hands and the payoffs for the made hands
and average value of the draw hands. If you know dealt variance, you
know drawn variance, assuming you already know total variance.
Multiplays only affect drawn variance.

wrote:

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

Also would anyone have the value of covariance for NSUD?

Even further could the formula for calculating the covariance of any
game be posted?

DWK

[deuceswild1000]

> Even further could the formula for calculating the covariance of

any

> game be posted?
>
> DWK

DWK

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

wrote:
> --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> <nightoftheiguana2000@y...> wrote:
> > Variance is a function of number of lines, for deuces:
> > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line

Var=~8

>
> So ARE YOU SAYING that the variance for five line play is
> significantly smaller than for single line? Approx. 8/26ths, even
> though the same amount of money is put at risk.

Yes.

I am not sure that I understand. Would you please show the
derivations of your calculations or some reference that I could study.

Your statement is not being discredited, it is just that there is not
enough explanation for me.

DWK

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "deuceswild1000"
<deuceswild1000@y...> wrote:

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
> wrote:
> > --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> > <nightoftheiguana2000@y...> wrote:
> > > Variance is a function of number of lines, for deuces:
> > > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line
Var=~8
> >
> > So ARE YOU SAYING that the variance for five line play is
> > significantly smaller than for single line? Approx. 8/26ths,

even

> > though the same amount of money is put at risk.
>
> Yes

http://wizardofodds.com/videopoker/vidpokapx3.html

Is this in agreement with you? If not, could your please
elaborate, as I do not know where your calculations come from.

DWK

I am not sure that I understand. Would you please show the
derivations of your calculations or some reference that I could

study.

Your statement is not being discredited, it is just that there is

not

···

> --- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>
enough explanation for me.

DWK

[nightoftheiguana2000]

My formula is for "Variance per bet":
3.140053 + 22.694565/N
Wizard's formula is for "Dollar Variance":
N x 3.140053 + 22.694565

These are both correct but Variance per bet is more useful. To get
"Dollar Variance" from "Variance per bet" you multiply by the amount
of the bet in dollars. If you are comparing a single line quarter
machine to a 5 play quarter machine, there is 5x more dollars per bet
on the 5 play - so, you multiply the Variance per bet by N (the number
of multiplays) to get dollar Variance. I.e.:
N x (3.140053 + 22.694565/N) = N x 3.140053 + 22.694565

Variance per bet is less on a multiplay, however you are betting more
per hand if you stick to the same base denomination, thus dollar
Variance is greater on a multiplay. For example, the general rule of
thumb is that each extra line requires an additional 10%, so 5 play
quarters will have 140% of the dollar swing of single play quarters.
On
the other hand 5 play nickels will have 140%/5=28% of the dollar swing
of single play quarters (i.e. variance per bet is reduced). For the
same bet, single line fpdw variance is 25.8 and for the same bet, 5
play fpdw variance is 7.7, the ratio is 7.7/25.8=30%, so the rule of
thumb above is a pretty good estimate. Of course if you're comparing
single line quarters to 5 play quarters, well, you're now betting 5x
per hand, so 30%x5=150%, again the rule of thumb above is a pretty
good estimate.

wrote:

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

--- In vpFREE@yahoogroups.com, "deuceswild1000"
<deuceswild1000@y...> wrote:
> --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> <nightoftheiguana2000@y...> wrote:
> > --- In vpFREE@yahoogroups.com, "deuceswild1000"
<deuceswild1000@y...>
> > wrote:
> > > --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> > > <nightoftheiguana2000@y...> wrote:
> > > > Variance is a function of number of lines, for deuces:
> > > > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line
> Var=~8
> > >
> > > So ARE YOU SAYING that the variance for five line play is
> > > significantly smaller than for single line? Approx. 8/26ths,
even
> > > though the same amount of money is put at risk.
> >
> > Yes

http://wizardofodds.com/videopoker/vidpokapx3.html

Is this in agreement with you? If not, could your please
elaborate, as I do not know where your calculations come from.

DWK

>
> I am not sure that I understand. Would you please show the
> derivations of your calculations or some reference that I could
study.
>
> Your statement is not being discredited, it is just that there is
not
> enough explanation for me.
>
> DWK

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

My formula is for "Variance per bet":
3.140053 + 22.694565/N
Wizard's formula is for "Dollar Variance":
N x 3.140053 + 22.694565

These are both correct but Variance per bet is more useful. To get
"Dollar Variance" from "Variance per bet" you multiply by the

amount

of the bet in dollars. If you are comparing a single line quarter
machine to a 5 play quarter machine, there is 5x more dollars per

bet

on the 5 play - so, you multiply the Variance per bet by N (the

number

of multiplays) to get dollar Variance. I.e.:
N x (3.140053 + 22.694565/N) = N x 3.140053 + 22.694565

Variance per bet is less on a multiplay, however you are betting

more

per hand if you stick to the same base denomination, thus dollar
Variance is greater on a multiplay. For example, the general rule

of

thumb is that each extra line requires an additional 10%, so 5 play
quarters will have 140% of the dollar swing of single play

quarters.

On
the other hand 5 play nickels will have 140%/5=28% of the dollar

swing

of single play quarters (i.e. variance per bet is reduced). For the
same bet, single line fpdw variance is 25.8 and for the same bet, 5
play fpdw variance is 7.7, the ratio is 7.7/25.8=30%, so the rule

of

thumb above is a pretty good estimate. Of course if you're

comparing

single line quarters to 5 play quarters, well, you're now betting

5x

per hand, so 30%x5=150%, again the rule of thumb above is a pretty
good estimate.

This sounds possible, but again, are there any references (hopefully
on line) that I could study.

WoO takes his std dev and multiplies by dollars times total hands
played. Again do not know if thes are compatible or not, but it
would seem that he gives both dollars bet and per single hand which
seems to be equal to your variance per bet.

Perhaps someone else might have some other way of explaining it.

DWK

[nightoftheiguana2000]

wrote:

This sounds possible, but again, are there any references (hopefully
on line) that I could study.

http://www.google.com/search?q=Variance

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

[Peter]

Dear NOTI,

In one of your earlier posts you say:

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:
> Variance is a function of number of lines, for deuces:
> Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line

Var=~8

On Jazbo's site, I read:

To compute the variance of an N-play version of a game from the
list, just add the base game variance to (N-1) times the covariance.

where he gives, for FPDW:

            Covariance of Various Games
Game EV Variance Covariance Correl. Coeff.

FPDW 100.762% 25.842 3.140 12.151%

So, according to Jazbo, for FPDW:

Var=25.842+3.140*(N-1)

You are saying that the difference in these two formule is that
yours is "Var per bet". His is "dollar variance". Do I have that
correct?

Can you direct us to a reference that shows how one goes from one to
the other, that is, "Variance per bet" to "dollar variance", or
return.

The jargon used here and when one moves from on prob & stat paper to
the next is not at all consistent. Also. Almost no papers I have
read deal with VP. Thus, my difficulty, untrained as I am, to
follow a lot of the points being made.

Thanks.

…..bl

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "bornloser1537" <bornloser1537@y...>
wrote:

Dear NOTI,

In one of your earlier posts you say:

> --- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
> <nightoftheiguana2000@y...> wrote:
> > Variance is a function of number of lines, for deuces:
> > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line
Var=~8

On Jazbo's site, I read:

To compute the variance of an N-play version of a game from the
list, just add the base game variance to (N-1) times the

covariance.

where he gives, for FPDW:

            Covariance of Various Games
Game EV Variance Covariance Correl. Coeff.

FPDW 100.762% 25.842 3.140 12.151%

So, according to Jazbo, for FPDW:

Var=25.842+3.140*(N-1)

I come to the same conclusion regarding the computation. It
appears as though WoO also has the same calculation.

You are saying that the difference in these two formule is that
yours is "Var per bet". His is "dollar variance". Do I have that
correct?

Bolh appear to multiply this number by the sqrt of the total number
of hands then times the dollars bet per bet to turn this into std
dev in dollars.

Can you direct us to a reference that shows how one goes from one

to

the other, that is, "Variance per bet" to "dollar variance", or
return.

Yes please

The jargon used here and when one moves from on prob & stat paper

to

the next is not at all consistent. Also. Almost no papers I have
read deal with VP. Thus, my difficulty, untrained as I am, to
follow a lot of the points being made.

I don't know if I put that in any of my post or not. I wrote it
once--may have lost it when I clicked on wrong bubble. I had to re-
read it because I thought I had written it:-)

Thanks.

…..bl

Thanks for concluding all of the same things I did also. If we are
both wrong, at least we are not alone.

Misery loves company

DWK

[nightoftheiguana2000]

wrote:

> <nightoftheiguana2000@y...> wrote:
> > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line
Var=~8
according to Jazbo, for FPDW:
Var=25.842+3.140*(N-1)

You are saying that the difference in these two formule is that
yours is "Var per bet". His is "dollar variance". Do I have that
correct?

Yes.

Can you direct us to a reference that shows how one goes from one

to

the other, that is, "Variance per bet" to "dollar variance", or
return.

No, but I can explain.

My formula is variance per bet, so for example, variance per bet is
the same on a 5 coin quarter machine as it is on a 10 coin quarter
machine as it is on a dollar machine for the same game, even though
the amount bet per play changes. To calculate the standard deviation
in dollars using variance per bet:
$sd=$bet/play x sqrt(plays x var)
for example, for 5-play nickels fpdw, 4900 plays:
var=3.14+22.702/N=7.6804
$sd=$1.25 x sqrt(4900 x 7.6804)=$242.49

The Jazbo formula for multiplay variance works as follows:
Jvar=25.842+3.140*(N-1)
for 5-play fpdw, Jvar=38.402
to get $sd, you use this formula:
$sd=$(base bet) x sqrt(hands x Jvar)
for example, for 5-play nickels fpdw, 4900 plays:
$sd=$.25 x sqrt(4900*5 x 38.402)=$242.49

···

--- In vpFREE@yahoogroups.com, "bornloser1537" <bornloser1537@y...>

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000"
<nightoftheiguana2000@y...> wrote:

wrote:
> > <nightoftheiguana2000@y...> wrote:
> > > Var=3.14+22.695/N, i.e. for single line Var=~26, for 5 line
> Var=~8
> according to Jazbo, for FPDW:
> Var=25.842+3.140*(N-1)
>
> You are saying that the difference in these two formule is that
> yours is "Var per bet". His is "dollar variance". Do I have

that

> correct?

Yes.

> Can you direct us to a reference that shows how one goes from one
to
> the other, that is, "Variance per bet" to "dollar variance", or
> return.

No, but I can explain.

My formula is variance per bet, so for example, variance per bet is
the same on a 5 coin quarter machine as it is on a 10 coin quarter
machine as it is on a dollar machine for the same game, even though
the amount bet per play changes. To calculate the standard

deviation

in dollars using variance per bet:
$sd=$bet/play x sqrt(plays x var)
for example, for 5-play nickels fpdw, 4900 plays:
var=3.14+22.702/N=7.6804
$sd=$1.25 x sqrt(4900 x 7.6804)=$242.49

The Jazbo formula for multiplay variance works as follows:
Jvar=25.842+3.140*(N-1)
for 5-play fpdw, Jvar=38.402
to get $sd, you use this formula:
$sd=$(base bet) x sqrt(hands x Jvar)
for example, for 5-play nickels fpdw, 4900 plays:
$sd=$.25 x sqrt(4900*5 x 38.402)=$242.49

How about using Wizard of Odds formula to do the same calc. If I
follow his cals I would get 5 times this answer or about $1212 for a
sd for five play.

http://wizardofodds.com/videopoker/vidpokapx3.html
http://wizardofodds.com/videopoker/vidpokapx3ans.html#q4

BTW I was willing to accept Jazbo's and WoO calc for covariance. I
assume from your other post that dealt variance and covariance are
equal in mutltplay but not in other games. IOW I would get $1.25
in place of $.25 in the above Jazbo formula. If you can resolve
this, I think we have finally come to an agreement.

BTW did you derive your formula or "learn" it from some published
material. Whether derived or learned, that is what I was asking
for. I just did not want to take something at face value without
supporting data,derivation, publicastion.

Like I pointd out earlier someone else saying it differently or
someone else asking it differently helps when two people cannot
communicate due to large difference in math abilities.

Thanks to you also BL

DWK

···

--- In vpFREE@yahoogroups.com, "bornloser1537" <bornloser1537@y...>

[nightoftheiguana2000]

wrote:

> The Jazbo formula for multiplay variance works as follows:
> Jvar=25.842+3.140*(N-1)
> for 5-play fpdw, Jvar=38.402
> to get $sd, you use this formula:
> $sd=$(base bet) x sqrt(hands x Jvar)
> for example, for 5-play nickels fpdw, 4900 plays:
> $sd=$.25 x sqrt(4900*5 x 38.402)=$242.49
How about using Wizard of Odds formula to do the same calc.

Same.

If I
follow his cals I would get 5 times this answer or about $1212 for

a

sd for five play.

http://wizardofodds.com/videopoker/vidpokapx3.html
http://wizardofodds.com/videopoker/vidpokapx3ans.html#q4

I don't think you did your calc correctly. In wizard's answer 4 cited
above, the dollar multiplier is $.25x5 i.e. $(base bet) for a ten play
quarter machine and the sd term is sqrt(10) i.e. sqrt(hands), same as
Jazbo formula above. Jazbo formula can be modified to $sd=$(base bet)
x sqrt(hands) x sqrt(Jvar) and sqrt(Jvar) would be J(standard
deviation) which is what wizard uses.

BTW did you derive your formula or "learn" it from some published
material. Whether derived or learned, that is what I was asking
for. I just did not want to take something at face value without
supporting data,derivation, publicastion.

Well, I learned it from a first year probability and statistics class.

···

--- In vpFREE@yahoogroups.com, "deuceswild1000" <deuceswild1000@y...>

[deuceswild1000]

--- In vpFREE@yahoogroups.com, "nightoftheiguana2000" <

I don't think you did your calc correctly. In wizard's answer 4

cited

above, the dollar multiplier is $.25x5 i.e. $(base bet) for a ten

play

quarter machine and the sd term is sqrt(10) i.e. sqrt(hands), same

as

Jazbo formula above. Jazbo formula can be modified to $sd=$(base

bet)

x sqrt(hands) x sqrt(Jvar) and sqrt(Jvar) would be J(standard
deviation) which is what wizard uses.

Well you finally convinced me. I think you may have learned that I
only needed some coaching and can understand this if I get some
help. Thanks.

So if I play FPDW single line at 25c, I have a std dev of $446. If
I play the same amount of dollars of 5c, five line I have a std dev
of $241.

Agree? If so, then why would I want to play the single line (your
take or any other readers also)?

> BTW did you derive your formula or "learn" it from some

published

> material. Whether derived or learned, that is what I was asking
> for. I just did not want to take something at face value

without

> supporting data,derivation, publicastion.

Well, I learned it from a first year probability and statistics

class.

Well my stat class were over 48 years ago, but I do not remember any
thing like this in them. But if it is published it should be able
to be referenced.

Keep up the good work. Just remember that some of us require more
help than others

DWK

[Peter]

Amen to that, brother!

A lot of us appreciate what NOTI provides.

Though, a little less of "... the next series of steps is left to
the reader ..." would be helpful. LOL!

Bridging the chasm between the formal mathematical principles of
probablility and statistics and how they should be properly appled
to VP is probably the most important input to VP understanding.
When many of us were in our first year probability and statistics
course, VP machines were not even a "gleam in the eye" of most
manufacturers and casino managers.

Thanks, again, NOTI!

.....bl

···

Keep up the good work. Just remember that some of us require more
help than others

DWK

[Peter]

So if I play FPDW single line at 25c, I have a std dev of $446. If
I play the same amount of dollars of 5c, five line I have a std dev
of $241.

Agree? If so, then why would I want to play the single line (your
take or any other readers also)?

DWK

That is certainly what the mathematics says. And, intuition seems to
back that up, with the point that, with the nickle 5-line, you are
playing more (i.e., 5 times as many) hands (with the same amount of
money) and thus you are "smoothing out" the highs and lows.

That is why we all try to "average" our VP playing time over our
entire lifetimes. The more hands one plays (with correct strategy, of
coourse) the closer one gets to the theoretical EV.

.....bl