# average & expected value

Discussion in 'Skilled Play - Card Counting, Advanced Strategies' started by sagefr0g, Aug 25, 2010.

1. ### sagefr0gWell-Known Member

can anyone give an explanation of the essential difference between a simple average (say of a huge set of data) and expected value?

if i know some average value, is there a way to view that as expected value or convert it to expected value?

edit: like ok, say i play some simple game, and i play it billions of times. say of all the money i make and lose that the average is \$8 from all those billions of plays.
could i say that \$8 is my expected value for that game?

2. ### assume_RWell-Known Member

From: http://en.wikipedia.org/wiki/Expected_value

"To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the arithmetic mean of the results. "

Essentially, to calculate the EV, you take each value (such as +\$2), and the probability of that occurring (such as 10%), and multiply (\$0.20). Then you take your next value (such as +\$3) and the probability of that occurring (such as 90%) and multiply (\$2.70). Adding all that up gives you the EV, or \$2.90.

Alternatively, you can run a billion simulations, and the mean will be very close to \$2.90. This is because about 900 million times you will win \$3 (so you get \$2.7 billion), and about 100 million times you will win \$2 (so you get \$0.2 billion). Your total money will be \$2.9 billion, and when you divide by your billion rounds (to get the mean, or average), you get \$2.90.

Does that explain it enough for you or was that confusing?

3. ### SuckerWell-Known Member

Yes. Your expected value IS your average. If you have a 1% advantage and bet \$100, your expected value is \$1 per bet, which means that you will average \$1 for every \$100 bet.

4. ### sagefr0gWell-Known Member

ahh, ok, thank you very much.

so, it's obvious you really do want to have billions of trials (at least in a lot of cases) so as to come up with the best value, sorta thing. while maybe keeping in mind the problem of standard error.

and so when it comes to variance, i believe one could get that from the raw data by taking the average of the squared differences from the mean, sorta thing?
then take the square root of that to get standard deviation?
again for most cases one would need a large number of trials, sorta thing.
and i guess this is all assuming that the data in question fits a normal distribution, bell curve, sorta thing?

but really i was just wondering if a simple average is pretty much the same thing as expected value, in the case i mentioned.
so are there situations where an average wouldn't fit the bill for expected value? errhh well i guess one case would be where not enough data is used to overcome standard error, maybe?

5. ### blackjack avengerWell-Known Member

There is EV and EV and EV

I think we most commonly look at EV per hour.
We can also look at it per hand.

However, as we know the variance of that one hand or one hr can be staggering.

Now, if one had an idea of the length of their career then one could come up with an EV for their career or perhaps many hours played, and this number would have more meaning because the variance would be less of a factor.

Perhaps EV per NO
or
EV per 4NO if you resize bank based on wins and losses

The devil is in the variance

:joker::whip:

6. ### psyduckWell-Known Member

This is confusing. I thought EV means advantage based on initial bet, not hourly win in \$.

7. ### SuckerWell-Known Member

You are exactly right. In order to obtain hourly win in \$, you have to multiply EV by hands per hour. Obviously, depending upon the dealer, how full the table is, whether or not you play one or two boxes, etc.; your hourly win rate will vary tremendously.

8. ### blackjack avengerWell-Known Member

A Bit Dated but Still Has it's Uses

Yes, it does mean that
but
before SCORE and score came about one would talk of hourly EV
Also, this would still have meaning if your bank is bigger then 10g or you don't bet optimally.
How much do you make an hour. What is your Expected Value for an hour's play or multiple hours or weekly, monthly or NO etc.

:joker::whip:

9. ### k_cWell-Known Member

If you sim a \$1 bet on each of 1 billion rounds and when done you are +\$8 then according to the sim EV in percent = 8/1,000,000,000*100 = .0000008% = ~0% = ~break even bet.

Ev is the amount in percent (or can be expressed as fraction) of bet that is expected to be added to (+EV) or subtracted from (-EV) bankroll as a result of one round.

However one must remember that the result for any given round = DidddlySquat(r), where DiddlySquat() is an array of unreliable numbers and r = round number. Assuming the sim is properly set up, though, EV can be obtained from the DiddlySquat array by summing the elements and dividing by total number of rounds.

For a 1 billion round sim:
EV = (DiddlySquat(1)+DiddlySquat(2)+.......+DiddlySquat(999999999)+DiddlySquat(1000000000))/1000000000

Instead of a sim EV could possibly be calculated with more accuracy using combinatorial analysis, thus eliminating the need for the DiddlySquat array.

:joker::laugh:

10. ### sagefr0gWell-Known Member

hmm ok, caught that on the part where you are saying what ev is, is how it is a expectation to add or subtract to the bankroll on a play by play sorta basis.

so that makes me think i didn't frame the situation correctly.
lemme try a whole new scenario, please.

like, say someone says to me that they know of a play for which over time as you make the play you will find that the average amount won is \$8 for a given play.
and let us say that person also claims to have tested this play billions of times.
so the question is, would i be correct in considering the expected value of that play to be ev = \$8 ?

Yes.

12. ### k_cWell-Known Member

\$8 is an abstract amount. I guess you're saying that on average a given play profits by \$8 per round tested over 1 billion rounds. However, how much is the bet per round? If the per round bet is \$1 then EV = 800% and after a billion rounds you will have earned \$8,000,000,000 on \$1,000,000,000 total action. That is you expect to earn \$8 for every \$1 wagered.

If \$1000 is the per round bet then EV = 0.8%. You will still have earned \$8,000,000,000 after a billion rounds but now your total action is 1,000,000,000 * \$1,000 = \$1,000,000,000,000. That is you will earn \$8 for every \$1,000 wagered.

EV is relative. If you multiply amount wagered by EV the result is the amount your bankroll is expected to change on average for 1 round in the long run.

Hopefully more clear.

13. ### sagefr0gWell-Known Member

yes, more clear, i believe i understand the distinction.
thank you.