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Probability

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Posted: #1
You may remember the "Monty Hall problem" from the movie "21".  There are several similar probability problems out there, such as: The Secretary Problem, Prisoner's Dilemma, or Matching pennies. I am a huge fan of probability theories.

We could all be more profitable gamblers if we knew a little more basic math and probability theory, I think.

Here are some links from wiki:

And, some that I posted in the "religion" thread...
 
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Posted: #2
OK, since no one wants to start, I will... 

I chose Kelly Criterion because I've read it before.  The basic premise is this: Kelly Criterion will tell you how much of your bankroll to wager on a specific football game/blackjack hand/IRA investment package.

I read the wiki and here's what I understand:

The formula is: f=(bp-q)/b

where,

 f = fraction of the current bankroll to wager
b = the odds received on a wager (which I calculate to be roughly 91% on a -110 wager)
p = probability of winning
q = probability of losing (exactly 1-p)

so, I inserted my known variables:

f = unknown (what we're going to solve for)
b = 91% or .91 (based on a -110 vig)
p = 56% or .56 (based on my football picks over the last several years)
q = 44% or .44 (1-p)

Immediately, I notice that Kelly doesn't allow for a push!   Hmmm... how do we account for this?

Anyway, f = (.91*.56 - .44)/.91 = 0.0764835 (or roughly 7.5%)

So, according to Kelly, I should be wagering roughly 7.5% of my roll on my first football game.  Then, after each wager, I should re-calculate my winning probability (based on my current record over a certain time frame, like the last 2 years plus this current season maybe?) and re-work this formula to determine what my next wager amount should be.

I'm no mathematician, and I admit that I am learning as I go here...  So, does this seem right?  It sounds to me like this will help gamblers to reduce their wager when they are cold and increase their wager when on a hot streak.  For years, I've just set my unit at the beginning of each season and kept it the same whether I was up or down until next season.  This seems to go against Kelly's ideas.

Thoughts?





 
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Posted: #3

I'll get back to this after the alcohol wears off.

 
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Posted: #4
Quote Originally Posted by kickinA:

I'll get back to this after the alcohol wears off.

  I was sitting at work bored.. so I had a little spare time to do mundane stuff like this....   But, now I'm off and my head hurts, so it's beer time for me as well!
 
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Posted: #5
Rox:
You make some excellent points.  When you break gambling theory down to it's basics, you begin to understand just how "stacked" the odds are against the average joe gambler.  I never understood gambling guys who don't think about these things... To me it never made any sense to gamble casually... If you're not serious, then you are a loser.

The two two pillars of success, as you pointed out, are the managing your bankroll and your winning / losing streaks...

1) ALWAYS protect your bankroll.

2) Gamblers MUST reduce wager when they are cold and increase their wager when on a hot streak.

Two simple and basic rules that apply to all types of gambling.  Yet how many times do you hear losers claiming they are "all-in on this one game" or "I'm chasing this team tonight".

Very good thread
~~~~~ZOSO~~~~~
 
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Posted: #6

It's all Greek to me.

 

 
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Posted: #7
I don't think people take their time calculating the probabilities before every bet they make... Otherwise the bookies would go bankrupt.
 
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Posted: #8

very good thread--to me it highlights the importance of NOT laying -110 juice when placing your bets.  if you use a sight like matchbook, your kelly criterion number starts out at 11.5% for your first wager--huge difference. 

 

kelly criterior premise ( i think ) is designed to 'keep you in the game' and to survive the swings that can bankrupt your bankroll.  if you truly do have a 56% picking rate then the law of large #s suggests that you will 100% become a millionaire but kelly criterion is designed so that you don't go to 0 before you can achieve ballerific status

 
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Posted: #9
Quote Originally Posted by WendysRox:

OK, since no one wants to start, I will... 

I chose Kelly Criterion because I've read it before.  The basic premise is this: Kelly Criterion will tell you how much of your bankroll to wager on a specific football game/blackjack hand/IRA investment package.

I read the wiki and here's what I understand:

The formula is: f=(bp-q)/b

where,

 f = fraction of the current bankroll to wager
b = the odds received on a wager (which I calculate to be roughly 91% on a -110 wager)
p = probability of winning
q = probability of losing (exactly 1-p)

so, I inserted my known variables:

f = unknown (what we're going to solve for)
b = 91% or .91 (based on a -110 vig)
p = 56% or .56 (based on my football picks over the last several years)
q = 44% or .44 (1-p)

Immediately, I notice that Kelly doesn't allow for a push!   Hmmm... how do we account for this?

Anyway, f = (.91*.56 - .44)/.91 = 0.0764835 (or roughly 7.5%)

So, according to Kelly, I should be wagering roughly 7.5% of my roll on my first football game.  Then, after each wager, I should re-calculate my winning probability (based on my current record over a certain time frame, like the last 2 years plus this current season maybe?) and re-work this formula to determine what my next wager amount should be.

I'm no mathematician, and I admit that I am learning as I go here...  So, does this seem right?  It sounds to me like this will help gamblers to reduce their wager when they are cold and increase their wager when on a hot streak.  For years, I've just set my unit at the beginning of each season and kept it the same whether I was up or down until next season.  This seems to go against Kelly's ideas.

Thoughts?







NEVER use full Kelly, but use a fraction of what the outcome is. So you cut your 7.5% by another fraction. Your 7.5% is when you know your probability is 100% correct (like a toss of a coin is 50%, roll of a dice is 1/6th). 7.5% for 56% gambler will send you broke quickley.

Your 56% is incorrect as well. The implied probability is on a game by game basis. Your record of 56% is not a true refelection, variance and regression to the mean will ensure that this is the case.
 
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Posted: #10
Quote Originally Posted by Socaddict:

I don't think people take their time calculating the probabilities before every bet they make... Otherwise the bookies would go bankrupt.


winners do

losers don't
 
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Posted: #11
I feel like I'm back in Calculus.....
 
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Posted: #12
23 people in a room.

What's the probability that at least two have the same birthday?


 
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Posted: #13
Quote Originally Posted by FarQue:



NEVER use full Kelly, but use a fraction of what the outcome is. So you cut your 7.5% by another fraction. Your 7.5% is when you know your probability is 100% correct (like a toss of a coin is 50%, roll of a dice is 1/6th). 7.5% for 56% gambler will send you broke quickley.

Your 56% is incorrect as well. The implied probability is on a game by game basis. Your record of 56% is not a true refelection, variance and regression to the mean will ensure that this is the case.


Good point.  The wiki article even mentions this:

"A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount decreases the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since overbetting is worse than underbetting, it makes sense to err on the side of caution and bet less than the Kelly amount."

Thanks for the input, everyone.  Let's keep this going and maybe by the time the HOF game comes along, we'll be better prepared for this next season!
 
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Posted: #14
Quote Originally Posted by tigerw05:

very good thread--to me it highlights the importance of NOT laying -110 juice when placing your bets.  if you use a sight like matchbook, your kelly criterion number starts out at 11.5% for your first wager--huge difference. 

 

kelly criterior premise ( i think ) is designed to 'keep you in the game' and to survive the swings that can bankrupt your bankroll.  if you truly do have a 56% picking rate then the law of large #s suggests that you will 100% become a millionaire but kelly criterion is designed so that you don't go to 0 before you can achieve ballerific status



Another great point.  I know Vanzack is a huge proponent of not laying -110 and this is just another proof of the value of that.
 
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Posted: #15
I like all of this but I am not sure if I understand it all completely.  Good post.  I have a lot of things to learn and I like that. 
 
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Posted: #16
What if you have an unlimitted bankroll?
 
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Posted: #17
Kazual: Thanks, man.  I feel the same way.  I am hoping that we can learn together.

Bluefin:  Then bet away, my friend.  Bet away.  ;)


..................................................................................

I've done a little more research today and discovered the concept of Expected Value (EV) versus Expected Growth (EG). 

In a nutshell, just because a bet has a positive expected value, it does not mean positive bankroll growth.  I've found a two-part article that addresses this concept.  Here are the links:  Article 1 and Article 2.

I don't know about you guys, but this stuff (although hard to grasp) really excites me.  I just hope it makes me a more profitable gambler.  Then again, how can it not?
 
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Posted: #18
Quote Originally Posted by FarQue:



NEVER use full Kelly, but use a fraction of what the outcome is. So you cut your 7.5% by another fraction. Your 7.5% is when you know your probability is 100% correct (like a toss of a coin is 50%, roll of a dice is 1/6th). 7.5% for 56% gambler will send you broke quickley.

Your 56% is incorrect as well. The implied probability is on a game by game basis. Your record of 56% is not a true refelection, variance and regression to the mean will ensure that this is the case.


I just re-read this and caught something that I missed on the first read...  while searching for a winning probability, I was lost as to what to use.  I defaulted to my capping ability (picking 56% winners last year). 

Although I use a database to produce picks, I've not yet had the intellectual tools to translate a confidence rating into an expected win probability.  This is probably another area where I need to expand my knowledge.  Thanks for the advice.

By the way, I have noticed in the forums that people say things like, "I have the Yankees winning this game 62% of the time."  Frankly, I believe most of these yokels don't have a clue what they're doing when producing this number.  But, I'm sure there are those that have a pretty reasonable method for coming up with this probability.  If you have any insight in to how to do this, I'd greatly appreciate it.
 
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Posted: #19
Quote Originally Posted by WendysRox:



I just re-read this and caught something that I missed on the first read...  while searching for a winning probability, I was lost as to what to use.  I defaulted to my capping ability (picking 56% winners last year). 

Although I use a database to produce picks, I've not yet had the intellectual tools to translate a confidence rating into an expected win probability.  This is probably another area where I need to expand my knowledge.  Thanks for the advice.

By the way, I have noticed in the forums that people say things like, "I have the Yankees winning this game 62% of the time."  Frankly, I believe most of these yokels don't have a clue what they're doing when producing this number.  But, I'm sure there are those that have a pretty reasonable method for coming up with this probability.  If you have any insight in to how to do this, I'd greatly appreciate it.


That is the million dollar question. There are a thousand and one ways to do this If you have an idea about it start with poisson and bayesian. The collection of sample data will be your first hurdle. What data to use and how to interpret it will be next. Or it could a simple formula, basically whatever works for you, just as long as you develop an edge

Your probability doesn't have to be 100% correct, just right more often then the market. We don't know the true probability of an event

Kelly is your staking plan, probability is another matter. You need a solid staking plan (kelly is the best) and an edge to win. You can't win without one of them
 
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Posted: #20

23 people in a room.

What's the probability that at least two have the same birthday?

 

Ahhhh Amarillo Slim... is it 50/50, and then with 30 people it hits like 70%+

 
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Posted: #21
Quote Originally Posted by Gambleholic:

23 people in a room.

What's the probability that at least two have the same birthday?

 

Ahhhh Amarillo Slim... is it 50/50, and then with 30 people it hits like 70%+?

By the way, this is a fucking awesome prop bet

 
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Posted: #22

This has the potential to be a really good thread, especially if we can bring in some other knowledgable people and keep the idiots out of it.

Nice job starting it!

I am no expert on Kelly (and I think other people have somewhat made the point I am going to make), but I think the big difficulty with it is that it's very difficult to assign a probability to a game. Sure, people like Van can do it, but most people will need to develop a method for calculating the probability of winning a game before they can use Kelly correctly. Once you have done that, using Kelly is not very difficult.

 
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Posted: #23
real quick... I've been studying this kelly, EV, Expected Growth, etc for a few days now...  I am beginning to understand all of the formulas (not quite, but almost). 

so, assuming that coming up with a win probability is all up to one's own criteria and therefore, can't be justified one way or another,, could someone at least check my math out here?

actual spread converted to win probability via ml = odds

projected spread converted to win probability via ml = probability

edge = (odds * probability) - 1

kelly = edge/(odds - 1)


example:

NYJ -3 @ Buf (actual)

NYJ -7.28 @ Buf (projected)

actual spread of -3 = ml of -156, therefore odds = 1.64

projected spread of -7.28 = ml of -397, therefore expected probability of winning = 79.88

edge = (odds * probability) -1 = (1.64 * .7988) - 1 = .31 or 31%

kelly = edge/(odds - 1) = .31 / (1.64 - 1) = .4843 or 48.43% of bankroll

?? Can this be right? If so, I'd have to reduce kelly by about 85% to feel safe doing this because I see 4 pt differences in projected spread and actual spread about 5-8 times a week!

.

 
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Posted: #24
   thanks for giving me something to do at work tomrrow  
 
 
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Posted: #25
I think I've messed up.  The odds I have seem incorrect.  Let's assume I don't shop for odds and I'm getting the normal -110 on my NYJ -3 bet.

ODDS = -110 or 1.91

And, I'm only going to win if NYG covers -3.  So, even though they might win the game 79.88% of the time, they will only cover the spread 63.61% of the time. I figured this by taking the difference between -3 and -7.28, which is -4.28.  By plugging 4.28 into a spread/ML conversion tool, I come up with 63.61% probability of winning.  I don't know if this is absolutely the best way to figure this.  But, it's better than what I said earlier, for sure.  And, with my limited knowledge, it's the best I can come up with right now...so...


p = 63.61% or .6361


Therefore...

edge = (odds * probability) -1 =  (1.91 * .6361) -1 = .2149 or

21.49% edge

kelly stake = edge/(odds - 1) = .2149 / (1.91 - 1) = .2361 or

23.61% of bankroll is Kelly stake

Now, almost everything I've read refers to playing with one-half Kelly stake.  1/2 of 23.61 is roughly 11.80, therefore, I will wager 11.80% of my bankroll on this play.

1/2 of Kelly = 11.80% or .1180



EV = (probability * Odds -1) * percentage of bankroll wagered =
(pO-1)*X = (.6361*1.91-1)*.1180 = .0253, or

EV = roughly 2.53%

EG = (((1+(O-1)*X)^p )* ((1-X)^1-p)) -1 =

(((1+(1.91 - 1) * .1180)^.6361)*((1-.1180)^(1-.6361))) - 1 =

(((1 + .91 * .1180)^.6361) * ((1 - .1180)^.3639)) - 1 =

(((1+.1073)^.6361 * (1-.1180)^.3639) -1 =
 
(((1.1073)^.6361 * (.8820)^.3639) -1 =

(1.0669*.9553) -1 =

1.019209 - 1 =  .019209 or...

1.92% of expected bankroll growth


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