Standard Deviation question

I would like to know the difference in the standard deviation for flat betting compared to if you were card counting.

I would like to know what the SD would be for flat bet $5 over 100 hands, 1,000 hands, and 10,000 hands. Can you provide it up to 3 standard deviation?

I would also like to now what the SD would be if the minimum bet was $5 with a maximum bet of $30 for card counting.

Can this be provided for the same number of hands of 100, 1,000, and 10,000?

I'm curious to see how much fluctation there really is between the 2 different styles of play.

I would imagine that the flat betting would have far less fluctions since the betting size never changes from $5.

Thank you.

"I would like to know the difference in the standard deviation for flat betting compared to if you were card counting."

Well, of course, you need to define bet spread, ramp, rules, etc., but I'll provide some guidance.

"I would like to know what the SD would be for flat bet $5 over 100 hands, 1,000 hands, and 10,000 hands. Can you provide it up to 3 standard deviation?"

For a rather common set of rules, the s.d. for a hand of blackjack is about 1.15 units. As s.d. is a square-root function, you take the number of hands played, then the square root of that number, and you multiply by the flat-bet size ($5), and then 1.15.

For 100 hands played, s.d would be 1.15 x $5 x sqrt(100) = $57.50. Naturally, 2 s.d.s. is simply double that amount, and three s.d.s, triple. So, how do you interpret and use these values?

Suppose you flat bet $5 and play BS with a -0.5% edge. For 100 hands ($500 total wager), your e.v. is to lose $2.50. So, with $57.50 s.d., you may make the following statements: 1) There is a 68.3% probability that your result will lie between -$60 and +$55 (subtracting and adding one s.d. to the e.v.); 2) There is a 95.4% probability that your result will lie between -$117.50 and +$112.50; and 3) There is a 99.7% probability that the result will lie between -$175 and +$170.

So, for "homework," you should be able to make similar statements for 1,000 hands and for 10,000 hands. Write back with your answers, if you would like us to check your work. :-)

"I would also like to now what the SD would be if the minimum bet was $5 with a maximum bet of $30 for card counting."

That depends on rules, ramp, etc., but, using simulation software, a typical result, with an optimal ramp for a 6-deck game, might yield one s.d. in the area of $15.50, so $155 per 100 hands played.

"Can this be provided for the same number of hands of 100, 1,000, and 10,000?"

Same method as above.

"I'm curious to see how much fluctuation there really is between the 2 different styles of play."

Well, you have to understand that you haven't even asked for the comparison between spreading $5 to $30 or flat-betting the AVERAGE BET of the 1-6 spread, which would be almost exactly $10. That would be a more reasonable question to ask. In that case, flat-bet s.d. would be double what we said from before (so, $115), while spread s.d. would be $155.

"I would imagine that the flat betting would have far less fluctuations since the betting size never changes from $5."

See above.

"Thank you."

My pleasure.

Don

Don,

Of all the nice people on this board, I thank you very much for responding. I haven't posted in a long time and you are the expert.

Below is some homework that I completed for the following:

SD up to 3 levels for $5 flat bet for 1,000 hands and 10,000 hands.

1.15 x $5 x sqrt (1,000). Sqrt of 1,000 is 31.62

1.15 x $5 x 31.62 = $181.80 or rounded up to $182.

2 SD is $364 and 3 SD is 546.

If -.5% for 1,000 hands ($5,000 total wagered) is -$25. SD1 will lie between -$207 and $157.

SD2 will produce a range of -$389 and $339.
SD3 will produce a range of -$571 and $521.

10,000 example is summarized below:

1.15 x $5 sqrt (10,000). Sqrt of 10,000 is 100

1.15 x $5 x 100 = $575. SD2 is $1,150. SD3 is $1,725

If -.5% for 10,000 hands ($50,000 total wagered) is -$250. SD1 will lie between -$825 and $325.

SD2 will produce a range of -$1,400 and $900.
SD3 will produce a range of -$1,975 and $1,475.

I wanted to see how frequently you would be in SD3. If SD2 is 95.4% and SD3 is 99.7%. Difference is 4.3%. For the left tail of the bell curve on he negative side, I would just take 1/2 of this, right? This would be 2.15%. So is it safe to say that 2.15% of the time you would have SD3.

Also, I was thinking what would be the probabilty of playing 2 - 100 hand sessions and both sessions being in SD3. Would basically just multiply 2.15% x 2.15% to get .046% right? I am just thinking worst case scenario.

Can you double check my work? Just wondering if it would make sense to increase flat be form like $5 to $10 if you wind up at SD3. Seems like it would swing back and you can make back your money. Just a thought. What do you think Don.

Much appreciated.

Brendan

Don,
As I lay down and try to get my 1.5 old son to bed, I had a couple of thoughts.

$5 flat bet example roughly produced +/- 12 units for SD1. Let's assume that we never exceed SD1 for playing 10 sessions of 100 hands/session. I'm just trying to make a point.

It is impossible to end up at -120 (10x-12) units if each and every session were -12 units for SD1. Heck the SD1 for 1,000 units is 36 units. This tells me that you have winning sessions obviously mixed in.

One would have to think that you couldn't have many multiple losing sessions in a row before you have winning ones to counter effect the losses, right?

Here is a question. Would you happen to know how many losing sessions in a row one could experience?

"Of all the nice people on this board, I thank you very much for responding. I haven't posted in a long time and you are the expert.

"Below is some homework that I completed for the following:

"SD up to 3 levels for $5 flat bet for 1,000 hands and 10,000 hands.

"1.15 x $5 x sqrt (1,000). Sqrt of 1,000 is 31.62

1.15 x $5 x 31.62 = $181.80 or rounded up to $182."

Right you are! Good start!

"2 SD is $364 and 3 SD is 546."

See? Nothing to it!

"If -.5% for 1,000 hands ($5,000 total wagered) is -$25. SD1 will lie between -$207 and $157."

A+.

"SD2 will produce a range of -$389 and $339.
"SD3 will produce a range of -$571 and $521."

You don't need me! :-)

"10,000 example is summarized below:

"1.15 x $5 sqrt (10,000). Sqrt of 10,000 is 100

1.15 x $5 x 100 = $575. SD2 is $1,150. SD3 is $1,725"

All good.

"If -.5% for 10,000 hands ($50,000 total wagered) is -$250. SD1 will lie between -$825 and $325.

"SD2 will produce a range of -$1,400 and $900.
"SD3 will produce a range of -$1,975 and $1,475."

Perfect.

"I wanted to see how frequently you would be in SD3. If SD2 is 95.4% and SD3 is 99.7%. Difference is 4.3%. For the left tail of the bell curve on he negative side, I would just take 1/2 of this, right? This would be 2.15%. So is it safe to say that 2.15% of the time you would have SD3."

No, not quite. First, the precise number for lower than two standard deviations is 2.28%. No matter, as we were just rounding. But, be careful how you express things. The total area below -2 s.d. is 2.28%. The total area below 3 s.d. is a very negligible 0.13%. So, the 2.15% that you quote above represents the precise (but rather unimportant and uninteresting) area that lies above -3 s.d. but below -2 s.d. It isn't a number that is often quoted or that is very useful.

"Also, I was thinking what would be the probability of playing two, 100-hand sessions and both sessions being in SD3. Would basically just multiply 2.15% x 2.15% to get .046% right? I am just thinking worst case scenario."

You have no idea what "worst-case scenario" can actually be!!! Again, with an understanding of the above explanation, you could multiply the two probabilities to express how unlikely it might be to have two consecutive 100-hand session losses of greater than 2 s.d.s. But, again, I'd certainly use 2.28% and not 2.15%. As small as the number may be, why would you expressly rule out the area to the left of 3 s.d.? That wouldn't make sense.

"Can you double check my work?"

See above. Nice job. Just be clear on our last point.

"Just wondering if it would make sense to increase flat bet from like $5 to $10 if you wind up at SD3. Seems like it would swing back and you can make back your money. Just a thought. What do you think Don."

Uh-oh! Your A+ grade grade just took a serious hit!! Very bad thinking. Basically, what you're saying is that, "BEFORE EITHER OF THEM HAPPENS, because two consecutive 2-s.d. losses are highly unlikely, if you experience one such loss, increase your bets for the next session, because that second loss, to make two in a row, is very unlikely." That second loss is just as unlikely as the first one was -- no more no less, BECAUSE the first one happened!

If you flip a fair coin and it lands heads nine times in a row, do you now bet tails because it is highly unlikely to get TEN heads in a row???? PLEASE, after all your good mathematical work above, do not go down this path!

"Much appreciated."

No problem. Just don't do anything silly.

Don

"As I lie down and try to get my 1.5-year-old son to bed, I had a couple of thoughts."

They are, unfortunately, not very good ones.

"$5 flat bet example roughly produced +/- 12 units for SD1. Let's assume that we never exceed SD1 for playing 10 sessions of 100 hands/session. I'm just trying to make a point."

I already know where you're going, and it's to a very bad place!

"It is impossible to end up at -120 (10x-12) units if each and every session were -12 units for SD1."

No it isn't, if you record them as separate sessions. You didn't say how much bankroll you started with, but losing 12 units ($60) ten times in a row is certainly not impossible mathematically. NEVER make the mistake of confusing "highly unlikely" with "impossible."

"Heck the SD1 for 1,000 units [sic: hands] is 36 units. This tells me that you have winning sessions obviously mixed in."

Maybe. Maybe not. If you're asking what the probability is of being 120 units ($600) behind flat-betting $5 for 1,000 hands, we can calculate that. Expectation is to lose $25, as we mentioned before. One s.d. is $182. So, to lose $600 is to lose, quite precisely, 3.16 s.d.s. So, while there's less than one chance in 1,000 that this would happen, do NOT call it "impossible."

"One would have to think that you couldn't have many multiple losing sessions in a row before you have winning ones to counter effect the losses, right?"

One would, unfortunately, be wrong. Play long enough and you will experience unspeakable losing streaks that will try your soul.

"Here is a question. Would you happen to know how many losing sessions in a row one could experience?"

Here is an answer: More than your bankroll could withstand, if you think about "doubling up" after losing sessions.

The economist John Maynard Keynes once famously said, while discussing brutal downturns in the stock market, "The market can stay irrational longer than you can stay solvent." Paraphrasing -- and please listen to me VERY carefully -- "If you aren't properly capitalized, you can experience seemingly irrational losing streaks that can and will last a lot longer than your (inadequate) bankroll."

A word to the wise. ...

Don

You know what was funny. Last night I thought about mentioning flipping a coin 10 times or using the example of a roulette wheel (less the 0 and double 0) as it relates to SD.

On the roulette wheel when you've seen it land on "red" ten times, you see everyone loading up on black along the way, only to get crushed as it continues landing on red.

I was wondering if each 100 hand session would be just like the roulette wheel. Basically each 100 hand session is independent set of results from the next. After a session, the game is reset again, right? Each spin of the roulettle wheel is exactly the same as a 100 hand session.

I guess you can have 10 lossing lessions in a row or more with blackjack even if you are playing perfect BS and card counting. You will various levels of SD all in the negative range. They will all vary from small negative SD to very large negative SD. Nevertheless, all of which are negative.

Don,
If SD is so detrimental to a blackjack player where it can literally wipe out a player due to the large negative swings, is there any possible way to use it to our advantage?

What I mean is if you were playing a game where the EV were 0%. The game would be exactly even. You would encounter just as many negative swings as positive swings, right?

How can a player somehow take advantage of the positive swings with the SD? Seems like we always get crushed by the negative swings. I take that back. We all lose due to the negative swings. Why can't we kill it on the positive swings. I guess if we knew this answer, blackjack would not be around for the casinos to offer, right?

All of this came about when my wife and I were playing at a small casino in lower part of California called Chumash based out of Santa Ynez. She was flat betting and never once raised her bet. We can only find a $10 table despite searching for a $5. Anway, I was varying my bets on positive counts by only very mildly. She would ask me for advice constantly on what to do. In the end she was up 21 units while I was down 6. Seems like she always does well. Basically she walked away every time when the table started to turn. Maybe that's why she one. I know cards don't have memory but there is nothing like walking away while you are up.

The flat betting "piqued" my interest. I remember you correcting me before on this word many months ago. I never knew how to spell it. Not sure if you remember. Thanks for everything. Your wealth of information is much appreciated.

"You know what was funny. Last night I thought about mentioning flipping a coin 10 times or using the example of a roulette wheel (less the 0 and double 0) as it relates to SD.

"On the roulette wheel when you've seen it land on 'red' ten times, you see everyone loading up on black along the way, only to get crushed as it continues landing on red.

"I was wondering if each 100 hand session would be just like the roulette wheel. Basically each 100 hand session is independent set of results from the next. After a session, the game is reset again, right? Each spin of the roulette wheel is exactly the same as a 100 hand session."

Now you're talking!

"I guess you can have 10 losing sessions in a row or more with blackjack even if you are playing perfect BS and card counting."

Yes, unfortunately, you can. Of course, if you're a slight favorite to win any given one-hour session (say 52-53%), then it is highly unlikely that you'll lose 10 in a row (.47^10 = 0.000526, or once in every 1,901 tries). So, I'm not saying that such bad luck is commonplace, but again, it isn't impossible.

"You will [see?] various levels of SD all in the negative range. They will all vary from small negative SD to very large negative SD. Nevertheless, all of which are negative."

Right.

"Don, If SD is so detrimental to a blackjack player where it can literally wipe out a player due to the large negative swings, is there any possible way to use it to our advantage?"

Short answer: no. Sorry.

"What I mean is if you were playing a game where the EV were 0%, the game would be exactly even. You would encounter just as many negative swings as positive swings, right?"

Right.

"How can a player somehow take advantage of the positive swings with the SD?"

Can't.

"Seems like we always get crushed by the negative swings. I take that back. We all lose due to the negative swings. Why can't we kill it on the positive swings. I guess if we knew this answer, blackjack would not be around for the casinos to offer, right?"

Right.

"All of this came about when my wife and I were playing at a small casino in lower part of California called Chumash based out of Santa Ynez. She was flat betting and never once raised her bet. We can only find a $10 table despite searching for a $5. Anyway, I was varying my bets on positive counts by only very mildly. She would ask me for advice constantly on what to do. In the end she was up 21 units while I was down 6."

Sucks, huh?

"Seems like she always does well."

"Seems" being the operative word.

"Basically she walked away every time when the table started to turn. Maybe that's why she won."

That's why she KEPT her winnings; it isn't why she won in the first place, instead of losing from the very first hand she played.

"I know cards don't have memory but there is nothing like walking away while you are up."

Yes there is -- walking away with even more! :-) Look, it's simple: if you have no edge, the more you play, the more likely it is that you lose. If you have the edge, just the reverse is true. But because s.d. dwarfs e.v. in the short run, anything can and will happen. Ploppies will win and card counters will lose. Just the nature of the beast.

"The flat betting 'piqued' my interest. I remember you correcting me before on this word many months ago. I never knew how to spell it."

Glad you remembered! :-)

"Not sure if you remember."

Yes, I do.

"Thanks for everything. Your wealth of information is much appreciated."

You're welcome. Trust the math, and shy away from the voodoo.

Good luck!

Don

Very nice tutorial Don.

Robbie