The Origin of 300 BB's

[Festus22]

Anyone know how this number was derived?

To put the question in context, let's say you're a winning player averaging 2 BB's/Hr and your hourly standard deviation is 2. IF your hourly rate is normally distributed, then 68% of all hourly results would be between 0 and +4 BB/Hr, 96% between -2 and +6 and 99% between -4 and +8. I believe this is incorrect based on practical experience. I also believe BB's/Hr is a poor baseline method because of the fact that an online player will have 50 - 60 sub-data points (hands) making up each hourly value while a B&M players might have 25. This gives the B&M player a higher hourly variance since each sub-data point has a greater influence on the mean and standard deviation.

With the above being said, it's almost inconveivable that if the hourly rate was even close to normal that one could have enough consecutive hourly results in the negative side of the expected hourly result that would add up to -300 BB's.

I'm not disputing the 300 number. I recently just suffered a -146 BB swing from my highest peak to the lowest valley and back - all within a span of about 1400 hands. I just would like to know how it was derived, what underlying assumptions were used and, in terms of expectation, how often one might expect a -100 or -200 swing using the 300 model.

[Nottom]

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[Festus22]

Thanks for the link.

Two questions I would have is A) How can that information be translated into a probability of hitting any chosen value instead of just a risk of ruin. Stating it another way, if I have hourly rate x, hourly standard deviation y and hours played z, what's the chance I'll lose 146 BB's (as in my case) or any other number.

And question B) I still don't get how hourly rate is a valid base. Isn't the choice of using an hour somewhat arbitrary? Why not per hand? Let's say I'm super-holdem-man and play 1000 hands/hr while my opposite plays 2. How can we both use the same hourly calculation and each have a valid result when superman has 1000 hands in his hour and opposite has 2?

I'm not Mr. megamath but it seems to me a time-based system can't be accurate for this very reason.

[Nottom]

You might want to ask this in the probability forum, since thats where all the Stat gurus like to hang out.

[pudley4]

Superholdem man will have a higher win rate/hr and higher standard deviation. These will offset, resulting in the same bankroll requirements and risk of ruin.

[magic_man]

Traditionally the value used for standard deviation is much higher. I don't have Gambling Theory and Other Topics with me right now, but I believe that standard deviation is normally taken to be between 8 and 12 bets/hr. Also, the 300 BB usually assumes a win rate of 1 bet/hr.

The equation for Risk of Ruin is:

RoR = exp(-2*u*BR/sigma^2)

u = win rate
BR = bankroll
sigma = std dev

Mason uses a different formula in his book for simplicity, but they give approximately the same results. If we then use sigma = 12 as the standard deviation, u = 1 for win rate, and BR = 300, we get:

RoR = 1.6%

If you want a bankroll for a specified RoR, use the following:

BR = [ln(RoR)]*(sigma^2)/(-2*u)

So for RoR = 1.00%, the required BR is 332 BB.

Basically the win rate is going to be somewhere between 1 and 2 (hopefully), and std dev should be from 8-12. Playing with these numbers gives bankrolls normally from 200-300 bets with low risk of ruin. Hope that made some sense.

~Magic_Man
"If the whole idea is not to show how it's done, how does ANYBODY ever learn card tricks?!"

[Homer]

Stating it another way, if I have hourly rate x, hourly standard deviation y and hours played z, what's the chance I'll lose 146 BB's (as in my case) or any other number.

Here is the short-term risk of ruin formula from Blackjack Attack. I see no reason why this formula cannot be applied here:

r = N((B-ev)/sd) + e^((2*ev*B)/var) * N((B+ev)/sd)

N() = cumulative normal distribution function
e = base of natural logarithm system (~2.7183)
B = short-term bankroll, expressed as negative number
ev = expected value
sd = standard deviation
var = variance (square of sd)

Example: Your hourly rate is 1 BB/hr, your hourly standard deviation is 10 BB/hr, you intend to play for 40 hrs and have a 146 BB bankroll. What is the probability of going broke?

B = 146
ev = 1*40 = 40
sd = 10*sqrt(40) = 63.25
var = 4000

r = N((-146-40)/63.25) + e^((2*40*-146)/4000) * N((-146+40)/63.25)

r = N(-2.94) + e^(-2.92) * N(-1.68)

r = .415%

-- Homer

[Homer]

And question B) I still don't get how hourly rate is a valid base. Isn't the choice of using an hour somewhat arbitrary? Why not per hand? Let's say I'm super-holdem-man and play 1000 hands/hr while my opposite plays 2. How can we both use the same hourly calculation and each have a valid result when superman has 1000 hands in his hour and opposite has 2?

Yes, using an hour is somewhat arbitrary. We use it because no one wants to know how much they can make per millisecond or per hand, they want to know how much they can make per hour.

As for your example, you will find that both players have the same EV/hand and SD/hand, and will thus have the same required bankroll to achieve a certain ROR. If you do it in terms of per hour, player 1 will make much more per hour, but will have a much higher hourly standard deviation. So it will work out that they both need the same bankroll. It just so happens that player 1 is playing the same game faster, so he'll make more money.

-- Homer

[Festus22]

I don't think I agree that someone playing 1000 hands/hr would have a higher hourly standard deviation than someone playing 2 - I believe it would be just the opposite. In the former case, you'd have 1000 individual hand results of which maybe 700 would be net 0 since you'd be folding PF for no bet and another 300 with either a positive or negative result. With that many points making up my result at the end of each hour, I would expect each hourly number to be pretty similar to the previous hour's result (normalized?). By only playing 2 hands/hr, you could have a huge swing in that a couple of hours might go by with net 0 when you fold and suddenly see a +20 result when you spike a hand. The SD in this case would be very high WHEN CALCULATED ON AN HOURLY BASIS.

I realize the above is an extreme example but I think it goes to the heart of my confusion of why I think a time based calculation is problematic. Do you think the formula you presented would work the same if you calculated everything on a per hand basis?

[Nottom]

[ QUOTE ]
I don't think I agree that someone playing 1000 hands/hr would have a higher hourly standard deviation than someone playing 2 - I believe it would be just the opposite

[/ QUOTE ]

On a per hand basis there should be little difference, but the SD of the 1000 hand/hour player is certainly going to be higher.

Think of it this way, in an extreme case the 2 hand/hour player might win 30BBs in his 2 hands so his SD is obviously going to be less than this. I would be shocked to see a playerkeep his SD under 30BB/1000 hands.

[Vehn]

Does that ROR formula assume that your winnings are being added to the bankroll so that your bankroll is constantly growing? If not, I really need to go blow some cash on something..

[Homer]

I don't think I agree that someone playing 1000 hands/hr would have a higher hourly standard deviation than someone playing 2 - I believe it would be just the opposite.

You are incorrect. If they have the same SD/hand, then the person playing 1000 hands/hr has a higher hourly standard deviation. Here's the formula you should be looking at:

SD/hr = SD/hand * sqrt(n), where n is number of hands played per hour

I think you are becoming confused because you know that in the long run your results become more reliable. You are thinking this means that your standard deviation goes down over time, but it does not. It goes down relative to your win rate. For example, say your EV is 1 BB/hr and your SD is 10 BB/hr. Your results look like this:

After 1 hr -- EV = 1 BB, SD = 10 BB's

- Your SD is 10 times your EV.
- You have a 54% chance of being ahead. (to find this in Excel you enter =1-normsdist(-.1))

After 10 hrs -- EV = 10 BB's, SD = 31.6 BB's (this is 10 * sqrt(10))

- Your SD is 3.16 times your EV.
- You have a 62.4% chance of being ahead.

After 100 hrs -- EV = 100 BB's, SD = 100 BB's (this is 10 * sqrt(100))

- Your SD is 1 times your EV.
- You have a 84.1% chance of being ahead.

So, you can see that over time your standard deviation continues to increase, yet your results become more reliable.

Do you think the formula you presented would work the same if you calculated everything on a per hand basis?

Yes, it works on a per hand basis. You just have to make sure that the EV and SD units are the same, be it per hand or per hour.

-- Homer

[Homer]

Does that ROR formula assume that your winnings are being added to the bankroll so that your bankroll is constantly growing?

I'm pretty sure the equation assumes that you don't touch the roll.

-- Homer

[Vehn]

If so then its pretty much pointless. As a semi pro I have a set bankroll amount of 333 bb that when I win over it, I "withdraw" the winnings and pocket them/blow them. If I have a loss I refill the roll before withdrawing again. I'd assume most of the internet players do the same thing. Whats our risk of ruin?

[Homer]

Whats our risk of ruin?

Zero. Since you reload, you have an infinite bankroll.

For a more mathematical answer, see BruceZ.

-- Homer

[Homer]

I think I might have been wrong about this. I'm not really sure.

[Vehn]

That's obviously not what I meant. I don't "reload" from outside sources, I refill it with poker winnings prior to withdrawing again.

[Homer]

Well, it wasn't "obvious" to me. I thought you were reloading with income outside of poker. No need to get testy with people who are trying to help. Whatever. Find your own answers.

-- Homer

[Festus22]

OK, I think I see what my problem is.

I was thinking that for my hourly SD, I just calculated the SD over all my hourly results. But per your formulas, that's not right.

Example: Let's say my hourly results for Mr. 1000 hands/hr are,

Hour 1 = +4 BB's
Hour 2 = -8
Hour 3 = +2
Hour 4 = +7
Hour 5 = -2

Now I take the SD of these and get 5.81. That's what I was thinking to use as my hourly SD.

In the case of 2 hands/hr, I might see something like:

Hour 1 = 0
Hour 2 = 0
Hour 3 = +22
Hour 4 = 0
Hour 5 = 0

In this case, the SD = 9.8.

So that's why I was thinking a time base was poor due to the much greater influence individual hand results could have on the hourly figure when so few hands were played per hour. I was not factoring in hands/hr.

I greatly appreciate your responses.

[Vehn]

I would hope you've read enough of my posts in the past two years to realize I'm not so utterly stupid to think that there is any value in attempting to calculate the risk of ruin for an infinite bankroll. But apparently I've either overestimated your common sense or underestimated my idiocy. My guess is the former based on your thoroughly condescending reply. The one from 3:09, not the one from 3:20.

[Homer]

I would hope you've read enough of my posts in the past two years to realize I'm not so utterly stupid to think that there is any value in attempting to calculate the risk of ruin for an infinite bankroll.

I was trying to answer the question I thought you had asked. I was making no judgement about your intelligence.

But apparently I've either overestimated your common sense or underestimated my idiocy. My guess is the former based on your thoroughly condescending reply.

It was in no way meant to be condescending. I don't respond to posts that way, regardless of whether the question seems difficult or trivial.

In case you hadn't noticed, I'm usually one of the first people to jump to the defense of those who are being attacked. What makes you think that I would choose to answer your question in a condescending manner?

-- Homer

[BillC]

1. The ROR equation assumes that you don't subtract from your BR.

2. Otoh, if you win say $100 then spend it on candy, you ROR is what it was to begin with. It is a function of your current bankroll. At B+100, your ROR is lower than at B.

I am sure of this, since I am a guru. Meditate on your navel until 1 and 2 don't seem contradictory. Spend time on the theory page.

[Nottom]

[ QUOTE ]
Example: Let's say my hourly results for Mr. 1000 hands/hr are,

Hour 1 = +4 BB's
Hour 2 = -8
Hour 3 = +2
Hour 4 = +7
Hour 5 = -2


[/ QUOTE ]

Do you really think these are reasonable estimates for Mr1000 hands?

I would really expect to see something more along the lines of.

Hour 1: -50BB
Hour 2: +127BB
Hour 3: +43BB
Hour 4: +6BB
Hour 5: -15BB

[Bozeman]

The only caveat with SD calculations is that you need enough measurements. Things like bankroll and RoR calculations use normal distributions, and the standard limit theorem says that the average of results from a non-normal distribution will approximate a normal probability distribution. For poker, results of one hand are not normal, but for 10-100 hands, they are very close to normal. So the RoR calculations are fine, except the short term one if you are using less than 10 hands.

Remember, SD is in units of x/sqrt(samples). Since samples (hands) is proportional to time, you can use 1/sqrt(hr). Suppose a standard SD is x in units of BB/sqrt(hand) (if standard SD is 10BB/sqrt(hr) for 40 hands/hr., x=5/sqrt(10)=1.58). For the 2 and 1000 hands per hour players, SD2= x/sqrt(hand)*sqrt(2hands/hr)=sqrt(2)x/sqrt(hr.)=2.24/sqrt(hr), and SD1000= x/sqrt(hand)*sqrt(1000hands/hr)=sqrt(1000)x/sqrt(hr.)=50/sqrt(hr).

Incidentally, the fact that these are proportional to 1/sqrt(time) means that it is misleading to speak of hourly SD which implies units of 1/time (that is, that SD scales with time played, when actually it scales with sqrt(time played)).



Craig

[Tommy Angelo]

"Why not per hand?"

I been saying that since back when floppies were actually floppy.

[JTG51]

Since I've been playing online more and have easy access to a hand count, I've been completely and totally won over to the per hand side. I'm curious though, how do you count how many hands you've played? Do you just make an estimate based on the number of hours you played or do you actually count each hand?

[Tommy Angelo]

"I'm curious though, how do you count how many hands you've played? Do you just make an estimate based on the number of hours you played or do you actually count each hand?"

I have gone through a couple stretches of 100 playing hours or so where I used chips in an abacus-like way to count exactly how many hands I was dealt, and how many of those times I saw the flop, also keeping track of freeplays from the big blind. This was done to satisty two curiosities: 1) I wanted to know what the numbers were 2) I wanted to know if I had the discipline to do it.

For the most part, I don't track anything except total net. And the way I have sometimes calculated that (for any specific duration) is to add up what I've spent, since my BR pretty much doesn't change. So I might not be the best guy to talk to about keeping records. :-) But I do find the "by-the-hand" concept most appealing.



Tommy

[JTG51]

I have gone through a couple stretches of 100 playing hours or so where I used chips in an abacus-like way to count exactly how many hands I was dealt, and how many of those times I saw the flop, also keeping track of freeplays from the big blind.

Before you responded I had a picture in my mind of you doing exactly that.

For the most part, I don't track anything except total net. And the way I have sometimes calculated that (for any specific duration) is to add up what I've spent, since my BR pretty much doesn't change. So I might not be the best guy to talk to about keeping records. :-) But I do find the "by-the-hand" concept most appealing.

LOL. Fair enough.

[magic_man]

As I understand it, this RoR formula gives your RoR at any given time with a certain BR. Therefore it would not assume that your bankroll is always growing. If you exceed the required BR for your desired RoR, you can spend that money freely and maintain a constant RoR. Some math genius correct me if I'm wrong please!

~Magic_Man
"If the whole idea is not to show how it's done, how does ANYBODY ever learn card tricks?!"

[CrackerZack]

Did I play with you in some SnGs yesterday?