Been trying to find the standard deviation for the game of Spanish 21. I take it the SD is dependent on rules so: H17, DDD 6Deck.
Anyone know where I can find the results?
SEE:
The Pro's Guide to Spanish 21 Appendix B.
Below -- Lost formatting, missing tables, etc.
We measure the fluctuation of our performance in a casino game,the risk factor, by its standard deviation. You might remember fromhigh school that the standard deviation of a sample of values is ameasure of its spread.
Example: I go to the casino every day for ten days and play100 hands per day of standard six-deck H17 using basic strategy,betting one unit per hand. My daily winnings are: ?3, ?22, ?10, 12,?5.5, 25, 6.5, 8, 7, ?10. The mean, or average, is the sum dividedby 10, which is 0.8. Therefore, my average result per day is a 0.8unit win, so I have been lucky. Obviously, this set of ten 100-handsessions is going to be different from another set of ten 100-handsessions. If we took the mean winnings of a million 100-handsessions, it would be ?0.78 units, because the house edge is0.78%, and we bet 100 hands of one unit each.
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We calculate the variance of these ten sessions by subtracting
each of the ten session results from the mean, squaring them, addingthe ten squares together, and dividing by ten. The standarddeviation is the square root of the variance. Our ten sessions have astandard deviation of 12.9 units. If we took another set of ten 100-hand sessions, we would get a slightly different standard deviation,as ten sessions is not sufficient to give us an accurate result. If wecalculated the standard deviation of a million 100-hand sessions, wewould obtain a figure very close to the true standard deviation of11.7 units. If you compare this figure to the expected return, ?0.78,you will notice just how much larger it is.
A large standard deviation is characteristic of all casino gamesand explains the luck factor, accounting for why an absolutebeginner with no knowledge of the game can occasionally make alot of money. However, as more and more hands are played, theexpected return decreases at a much faster rate than the standarddeviation increases and eventually overtakes it. This is why it ispossible for the regular player to win in the short term, but not in thelong term. The short-term wins keep the player coming back.Overall, the player will lose slightly more often than he wins, so thenet long-term effect is a substantial loss. Attempts to regain thismoney will ultimately result in even greater long-term losses.
If we graph a player’s cumulative winnings versus time, it willshow a “random walk with a downward drift.” The standarddeviation is proportional to the average amplitude of the randomwalk, and the house edge is proportional to the downward drift.For a picture of a random walk with an upward drift, which is whatthe advantage player experiences, see Figure 8.1.
The standard deviation can tell us a lot about the results that we
can expect over a particular time frame. The range is defined as sixtimes the standard deviation, and since the standard deviation for100 one-unit hands is 11.7 units, we know that if we go to thecasino tomorrow and play 100 hands, we will win somewherebetween 0.78 ? (3 × 11.7) and ?0.78 + (3 × 11.7), i.e., ?35.9 and34.3 units. (There is still a 0.1% chance that we will lose more than35.9 units, and a 0.1% chance that we will win more than 34.3units.) If our average bet is $10, we will win somewhere between?$359 and $343. Hence, the standard deviation quantifies the best-and worst-case scenario and is the standard measure of risk. Youwould already know that if you bet $10 per hand, you’re not goingto walk away with $5,000 after a day of playing; time and timeagain, your playing results will confirm the notion that your potentialwinnings and losses are constrained by the standard deviation. Luckcan be quantified.
The conventional symbol for standard deviation is a, the lower-case Greek letter sigma. (You may be familiar with ?, which isupper-case sigma.) In algebra, we can eliminate the multiplicationsign.
In general, the standard deviation for Blackjack, flat betting bunits per hand, is 1.1b?n , where n = the number of hands.Substituting n = 1, we get the standard deviation for one hand, ?l =1.1b, so the standard deviation forn hands, ?n= ?1?n. Thestandard deviation is equal to the single-hand standard deviationmultiplied by the square root of the number of hands. This is aproperty of all casino games. If b = 1, i.e., we bet one unit, ?l =1.1. This figure is sometimes referred to as the volatility index (VI).Each game has its own VI.
The volatility index of 1.1 for Blackjack is rounded to onedecimal place. If we look at the figures for specific Blackjack rules,we find that it varies. Table 4.11 shows the standard deviation forvarious Blackjack and Spanish 21 games. Doubling and splittingincrease it, because they increase the potential winnings and losses;redoubling increases it even more. Forfeiting, surrendering, OBBO,BB+1, SPL1, SPL2, and last-chance doubling decrease thestandard deviation.
The standard deviation for Blackjack is slightly less than NorthAmerican Spanish 21, and about the same as Pontoon. In Spanish21, we double more often, since doubling is allowed on any numberof cards, and we split 20% more often. However, in Pontoon, thisis totally compensated by the many variance-lowering rules: nosplitting to more than three hands and, except for Pontoon 1 and 7,no resplitting Aces; no doubling on soft hands; OBBO and BB+1.Pontoon 6 has the lowest standard deviation because we doubleless often, due to both last-chance doubling and the lack of OBBOor BB+1.
Table 4.11 Standard deviation for BJ and SP21
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There is no significant difference in the standard deviationsbetween H17 and S17, or between the six- and eight-deck games.When considered individually, NS and NDSA have no significantimpact on the standard deviation. Note that all of the standarddeviation figures I quote are for basic strategy. Obviously, ifsomeone plays suboptimally by never doubling or splitting, he willget a much lower standard deviation but at a huge cost.
Let’s suppose we play a one-hour session of six-deck H17 at$15 per bet, playing at a rate of 80 hands per hour. Our expectedreturn, or mean, after one hour is $15 × 80 × ?0.0078 = ?$9.36.Our standard deviation is 1.17 × ?80 × $15 = $157. If we played100,000 such sessions and plotted a histogram of our results fromeach session rounded to the nearest dollar, and joined the points onour graph together, they would give us a bell-shaped curve knownas the normal distribution, shown in Figure 4.2. (A histogram is agraph where the horizontal axis represents events and the vertical
axis corresponds to the frequency or number of times that theevents occur.) We would find that 68.2% of our results would fallwithin one standard deviation of the mean (34.1% each side),27.2% would fall between one and two standard deviations of themean (13.6% each side), 4.3% would fall between two and threestandard deviations of the mean (2.1% each side, rounded to onedecimal place), and 0.3% would fall outside three standarddeviations of the mean (0.1% each side, rounded to one decimalplace). You should be able to see that the areas under thecorresponding sections of the bell curve are proportional to thesevalues. Therefore, if we played 1,000 one-hour sessions, we wouldexpect to win more than ?$9.36 + (3 × $156.97) = $462 in onlyone or two sessions (as the figure of 0.1% is rounded down).
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Figure 4.2 The standard normal distribution
The expected loss for nine hours of play is $84.24, nine timesthe expected hourly loss. In mathematics, we call this a linearrelationship. However, the standard deviation for nine hours is 1.17×$15×’‘i(9×80)=3×(1.17×$15×J80),whic hisonlythreetimes the one-hour standard deviation. Similarly, the standarddeviation for a 36-hour week of play is only six times the hourlystandard deviation. Technically speaking, it is an inverse-quadraticrelationship rather than a linear one. The ratio of the standarddeviation to the expected return for a particular time frame isproportional to 1/?n, where n = number of hands, i.e., it decreasesin magnitude as the time frame increases.
When we are an advantage player, our expected return isgreater than zero. Like a basic strategy player, the more hands weplay, the more we reduce our standard deviation relative to ourexpected return, i.e., our relative risk. If our playing advantage was1%, which is a typical Blackjack counter’s advantage, and weplayed 1,000 hands with an average original bet of $25, ourexpected return would be 1% × $25 × 1,000 = $250, and ourstandard deviation would be 11,000 = 31.6 times the standarddeviation for one hand, ?l. If we bet ten times as much, i.e., $250per hand, we would make that $250 in only 100 hands, but ourstandard deviation would be 10 × -4100 = 100 times as much as?1. The more hands we play, the more we reduce our risk relativeto our expected earnings. This is one of the reasons why Blackjackplayers often play in teams, with one team member on each table.Ten players can play ten times as many hands as one player and canshare a combined bankroll.
Last edited by ZenMaster_Flash; 08-27-2016 at 06:39 PM.
Notes:
Re-doubles are always capped at the table maximum bet.
"Doubling for less" on a re-double may be forbidden;
at least when the "extra" money is truly minimal.
There must be a critical fraction (unknown) as to where a
'double for less' to "reach" the table max makes sense.
If you have a multi-card 11 facing a dealer stiff what is
the crucial fraction? If I have $400 at stake at a $500
max table, is it correct to 'double' by getting a card for
a $100 increment to my bet? I certainly count that.
It is plainly obvious that for a really small amount
it cannot be correct, but for an amount close to the
already doubled bet, it will be correct. To further
muddy the issue, it would vary by the fraction, the
hand matchup, and the expectation by True Count.
Doubling for less in order to play a redouble on a hand
of 12 is forbidden. Having said that, I would hope to
"get away with it" anyway.
Yes there are many factors which affect the proper strategy and thus, the requested SD. Because the amount of the initial bet itself is a factor, there are an infinite number of theoretical SDs and certainly an extremely large number of practical ones. I chose to list one reasonable choice which was for an initial bet of $25 which maximizes EV with respect to the super bonus and requires only a table max of only $100 when the size of the final double is restricted to table max, or a table max of $200 when the size of the total bet is restricted.
Double for less was ignored for the reasons that you stated, although I can compute the required critical fractions. Another factor affecting the strategy for re-double for less is the procedure for handing rescue. In most cases you keep the amount of the previous double and lose the rest.
I also want to clarify your commend regarding the role of the critical fraction when doubling vs re-doubling.
- For the initial double, we always want to double for the full amount and should not double unless we can exceed the critical fraction of our initial bet.
- When re-doubling, there are cases where we also want to re-double for the full amount and should not do so unless we can exceed the critical fraction. However, when re-double for less is correct, we want to double for as little as possible (0 would be ideal, but not practical). In these cases we do not re-double unless we can do so for less than the critical fraction.
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