Standard Deviation and Variance

**knoxstrong** · 01-31-2018, 01:32 PM

I came across a blackjack table yesterday with a surprisingly lucrative side bet opportunity. The bet is for a player total of 21 (blackjack) on the first two cards only. The payout is 19:1. As we all know, the average blackjack occurs approximately once every 21 hands, so the house has a nice edge off the top. However, I can already tell this bet is very lucrative by using an appropriate counting system. This side bet can be made ANYTIME throughout the 6-deck shoe. Pen is at slightly below 5 out of 6 decks (call it 4.8). I intend to attack this side bet as follows:

1. Abandon my standard blackjack count and focus ONLY on the side bet system. Here are my tag values for the side bet: Ace through 10s: -4, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1.
2. I will simply bet the table minimum on blackjack and play according to basic strategy, and then bet on the 21 side bet when the TC is 1 or higher.

I am already convinced of the very high EV this will produce. However, I'm not certain of the deviations/variance I can expect with this game. I expect this side bet will have a very high variance factor given that the bet will only be won, theoretically, once every 15 hands or so during good counts. The variance will have a large impact on the required bankroll to beat this game. I am not adept at determining SD and variance on new games like this. If anyone can show me how to accomplish this I would be greatly appreciative.

**DSchles** · 01-31-2018, 08:58 PM

You don't need s.d. and variance. You need to know the correct e.v. for the side bets you'll be making. And then, because the payout of a winning bet to a losing one is 19:1, the precisely correct Kelly wager is to bet (bankroll * e.v.)/19.

Don

**knoxstrong** · 01-31-2018, 09:23 PM

The maximum on the side bet is $10. The EV is 4.94 units per 100 hands. How much bankroll is needed to obtain a zero percent risk of ruin?

**Meistro123** · 01-31-2018, 10:55 PM

most counts are pretty good at predicting blackjacks so I don't see why you would use a specialized count for this sidebet

**weballinoutacontro** · 01-31-2018, 11:07 PM

Originally Posted by Three

What happened to the just answer the guys question and then shut up mentality. I guess you just have a double standard. LoL

Don's answer was directly on point with the information given.

As for the standard deviation of this game, SD=sqrt E(X^2)-E(X)^2, where E represents expected value function. Proof for this is below:

Simply substituting the numbers from the OP to obtain variance, according to OP's estimate of a 1/21 probability of BJ paying 19:1, results:
=((19^2)*20)/21 - ((19*20)/21)^2=111.61=Variance, taking the square root, SD=10.6

Obviously, though, this doesn't account for the effects of counting. I don't know BJ probabilities off the top of my head, but you could figure this at any True Count if u have that data

If you want to calculate RoR, simply substitute into this formula:

where

and mu is the mean, and s is your bankroll.

Note that a zero RoR is physically impossible. By zero you probably actually mean something like 1%, or 0.1%.

**Bubbles** · 02-01-2018, 12:28 AM

Originally Posted by knoxstrong

How much bankroll is needed to obtain a zero percent risk of ruin?

An infinite bankroll.

I found a similar side bet on a single deck. I wasn't sure the pivot, but I had an extremely high count with all 4 aces left, so I decided to bet it. Apparently they only let you bet it on the first round. No counting that side bet :-(.

Sent from my SM-G955U using Tapatalk

**ZenMaster_Flash** · 02-01-2018, 08:55 AM

Three, chill out.

I, for one, felt that Don's post was excellent.

A good teacher redirects the student when the
student is certainly asking the wrong question.

This is "Socratic Questioning" and it serves
to remind us of Don's pedagogical prowess.

Thank you Don. Sincerely. ZMF

"You don't need s.d. and variance.
You need to know the correct e.v.
for the sidebets you'll be making."

**knoxstrong** · 02-01-2018, 10:22 AM

Ok all, I see the error of my ways. Thank you for the guidance. Here are some more adequate details. Please keep in mind that the standard blackjack hand is not being included in these values, nor should it be. As for the side bet, here is what we are looking at:

1. Maximum Wager - $10 (I will bet the max every time, because the count will be in my favor when I do place the wager)
2. EV on the Wager - 13.81% (or $1.38 per wager based on the $10 wager)

Since there I no point in calculating the Kelly wager, I'm assuming that I can back calculate the appropriate bankroll for this side bet by using Don's formula: Kelly Wager = (bankroll*EV)/19

Therefore, with EV stated as a decimal (0.1381), and the wager of $10 inserted as "Kelly Wager", the necessary bankroll to weather the swings on this side bet will be as follows:

$10 = (bankroll*0.1381)/19
$190 = bankroll *0.1381
$1375.81 = bankroll

How is my aim? This appears logical to me.

**Meistro123** · 02-01-2018, 12:20 PM

actually there is a point in calculating the wager, and you should be trying to determine your edge per true count in order to do so, not blanketly assigning a random? value

**weballinoutacontro** · 02-01-2018, 12:20 PM

Originally Posted by knoxstrong

How is my aim? This appears logical to me.

No. The bankroll variable in that Kelly formula is not "bankroll required to have tiny RoR." Kelly is about bet sizing, and achieves RoR=0 through constantly resizing bets.

If you want to calculate RoR, stick those numbers into the formula I provided on the last page. I suspect a $1376 BR will have a very high RoR

**knoxstrong** · 02-01-2018, 02:24 PM

Thanks weballinoutacontro. If I have understood you correctly, the only two formulas I really need for my purposes are the P(ruin) and the r = etc. that you previously posted. If that is the case, I have a couple questions:

1. The "u" or "mu" as you called it appears in both formulas. You defined it as the "mean". Which mean are you referring to?
2. I'm lost on what the "O" character refers to in the second formula. What am I plugging in there?

If you would prefer to not provide any further explanation, which I would understand, I can probably review the details at the source from which these formulas were collected. What is your source?

Thanks again to you and everyone who has responded on this thread. I want to attack this game, but need to ensure I'm using an appropriate bankroll.

**Gramazeka** · 02-01-2018, 06:01 PM

https://www.blackjackinfo.com/commun...52/#post-11980

**weballinoutacontro** · 02-01-2018, 06:18 PM

Originally Posted by knoxstrong

Thanks weballinoutacontro. If I have understood you correctly, the only two formulas I really need for my purposes are the P(ruin) and the r = etc. that you previously posted. If that is the case, I have a couple questions:

1. The "u" or "mu" as you called it appears in both formulas. You defined it as the "mean". Which mean are you referring to?
2. I'm lost on what the "O" character refers to in the second formula. What am I plugging in there?

If you would prefer to not provide any further explanation, which I would understand, I can probably review the details at the source from which these formulas were collected. What is your source?

Thanks again to you and everyone who has responded on this thread. I want to attack this game, but need to ensure I'm using an appropriate bankroll.

The source is simply Wikipedia: https://en.wikipedia.org/wiki/Standard_deviation
https://en.wikipedia.org/wiki/Risk_of_ruin

The "O" is a sigma, which represents standard deviation. Variance is sigma squared (i.e., the SD squared). The mean, mu, is the mean of your distribution