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EV and variance...the meaning of mean.

Average is used interchangeable with the word mean, but is NOT the same
as median and mode. Mean, Median, and Mode are all measures of the
central tendency of a probability distribution, but their definitions
are precise and different from one another. There are two types of
mean, arithmetic mean and geometric mean.

Yes, in everyday usage, the word "average" commonly refers to the mean, but not always. Consider the "average family." The "mean" is the sum of all values, divided by the number of such values. How can you apply values to families to be able to calculate a mean? You can easily calculate the mean number of children in a family, but that does not define an "average family." In this case, the mode (the most frequently occurring) is probably closer to what is meant by "average."

So as I said, the word "average" is not well defined, especially in mathematical terms. Expected Value (EV) is a kind of average, but strictly speaking it is not a mean because it is weighted by probabilities.

>>> In some cases, two possible holds with similar EV occur in the same
dealt hand, and the best hold (highest EV) may be opposite of when each
of these draws occurs by itself. Dan Paymar<<<

Can you explain in more detail what you are talking about in the above
statement.

Consider the following dealt hands in standard full pay Deuces Wild:
4c-5c-Kd-8c-Qh Best EV is 0.35523 for 4c-5c-8c (SF3di)
4c-5c-6d-8s-Qh Best EV is 0.34043 for 4c-5c-6d-8s (Inside straight)

When considering these possibilities separately, the double inside straight flush is clearly better than an inside straight draw, so we would tend to put the SF3 di above the Straight 4i in a hand rank table. Now consider the dealt hand:

4c-5c-6d-8c-Qh

Here we have a choice between those same two draws, and it turns out that the EV is 0.34043 for either 4c-5c-8c or 4c-5c-6d-8c. Is there any reason to prefer either of these possible holds over the other? The EV by itself says there's no difference, but if you're really serious about the best play then the inside straight draw is better because the variance is lower.

A recreational player probably should not be concerned about this, and I would have no problem with always putting the SF3 di above the Straight 4i in the strategy table, but the difference in long term variance is not insignificant. If you are concerned about penalty cards, then you should also be concerned about situations such as this.

Dan

···

"adv_vp" <adv_vp@yahoo.com> wrote:

--
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

The mean and average outcome are mathematically the same and both
refer to the expectation of a random variable. One must be careful
when mixing the ideas of sampling and distributional measurements.
Only when talking about sampling is it true that "The "mean" is
the sum of all values, divided by the number of such values". If one
has the probability distribution such as in vp then the mean or
average outcome is the weighted sum of payoffs times associated
probabilities. In fact if one samples repeatedly, as in simulations,
then the sum of all the outcomes divied by the sample size converges
to the mean or ER. This is the weak law of large numbers.
Dale

"adv_vp" <adv_vp@y...> wrote:
>Average is used interchangeable with the word mean, but is NOT

the same

>as median and mode. Mean, Median, and Mode are all measures of the
>central tendency of a probability distribution, but their

definitions

>are precise and different from one another. There are two types of
>mean, arithmetic mean and geometric mean.

Yes, in everyday usage, the word "average" commonly refers to the
mean, but not always. Consider the "average family." The "mean" is
the sum of all values, divided by the number of such values. How

can

you apply values to families to be able to calculate a mean? You

can

easily calculate the mean number of children in a family, but that
does not define an "average family." In this case, the mode (the

most

frequently occurring) is probably closer to what is meant by
"average."

So as I said, the word "average" is not well defined, especially

in

mathematical terms. Expected Value (EV) is a kind of average, but
strictly speaking it is not a mean because it is weighted by
probabilities.

> >>> In some cases, two possible holds with similar EV occur in

the same

>dealt hand, and the best hold (highest EV) may be opposite of

when each

>of these draws occurs by itself. Dan Paymar<<<
>
>Can you explain in more detail what you are talking about in the

above

>statement.

Consider the following dealt hands in standard full pay Deuces

Wild:

4c-5c-Kd-8c-Qh Best EV is 0.35523 for 4c-5c-8c (SF3di)
4c-5c-6d-8s-Qh Best EV is 0.34043 for 4c-5c-6d-8s (Inside

straight)

When considering these possibilities separately, the double inside
straight flush is clearly better than an inside straight draw, so

we

would tend to put the SF3 di above the Straight 4i in a hand rank
table. Now consider the dealt hand:

4c-5c-6d-8c-Qh

Here we have a choice between those same two draws, and it turns

out

that the EV is 0.34043 for either 4c-5c-8c or 4c-5c-6d-8c. Is

there

any reason to prefer either of these possible holds over the

other?

The EV by itself says there's no difference, but if you're really
serious about the best play then the inside straight draw is

better

because the variance is lower.

A recreational player probably should not be concerned about this,
and I would have no problem with always putting the SF3 di above

the

···

--- In vpFREE@yahoogroups.com, Dan Paymar <Dan@O...> wrote:

Straight 4i in the strategy table, but the difference in long term
variance is not insignificant. If you are concerned about penalty
cards, then you should also be concerned about situations such as
this.

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