Delicacy of Winning/Losing streaks in Betting

[uva3021]

The last post didn't garner much response, aside from the expected comment from Irish Tim, maybe people are hesitant to ask questions or feel the mathematics is over their head. Trust me its not, because I'm not that smart and Google is more than enough to introduce you to some of the basic concepts of probability and statistics. After exploring and doing some reading, one realizes the professionals in sports gambling are as resistant to dispersion of intelligence factor as any other industry, and if you want to really be successful, you have to do the research and apply yourself. So again don't be afraid to ask questions or comment, and I'll try to answer as best I can by in all honesty using google or some other source.

I’m briefly going to elucidate the delicate conditions surrounding a normal wager. Normal in this case being a standard issue spread where both sides are -110, regardless of the actual spread number. This is basically an extension of my expected value post, so as to build a continuum of relevant information that one can reference to in the future.


When I say briefly, I of course mean devoid of all the actual proofs and sophisticated mathematics that go into the manifestation of the essential tools that are employed. Because I am not that smart, and thankfully others that are more well versed and studied on the topic have provided freely the methods on the internet.


Let’s return to a common situation where the probability of success in a dichotomous scenario is 50%. In any universe there is only one of two possible outcomes. The expected number of success over N number of trials is of course the probability multiplied by N. Where p is the probability of success, the number of expected successes is measured as:


S = pN
For example:
p = 50%
N = 100
S = 50


This correlates to an expected value, explained here, that remains constant given N.


How fragile is that expected value when we are dealing with standard overrounds (remember EV is 5% loss for every -110 wager)?


To assess the degree of fragility in this state, one can assume fragility can also indicate streaks. Fragility has more meaning with dealing with expected growth as opposed to expected value, which like I said I will explain at length. Regardless, fragility still applies to expected value over N number of games with the same variables.


Imagine losing three games in a row after betting just ten games, using either intuition, a sophisticated and complex modeling system, or however else one ventures to evaluate possible bets. Obviously, such a streak, after only a small sample of ten games, one becomes irritated and seeks alternative methods to meet some unrealized and probably unrealistic goals. It is almost impossible to control our emotions, even though emotions can be counter intuitive in trying to achieve long-term goals.
In order to combat an overreaction to losing or even winning streak, the probability of such losing or winning streak must be calculated. This is necessary to put things into perspective, to compare what is expected to what is anomalous.


I’m going to just use the three game winning streak as the main explanatory note here


First, what are the odds of one team losing three games in a row?
This is simple, just a straight multiplication of the odds:


50% * 50% * 50% = 12.5%


Now let’s add another dynamic to the scenario. What are the odds a team loses at least three games in a string of four contests?


All the permutations of the two possible outcomes over a span of four games have to be computed. In terms of Winning and Losing:


LLLL, LLLW, LLWW, LWWW, LWLW, LLWL, LWLL, LWWL, WWWW, WWWL, WWLL, WLLL, WLWL, WLWW, WWLW, WLLW


16 possible combinations of wins and losses (this is easily double checked by multiplying 16 times .50^4), but the concern here is losing AT LEAST three games.


LLLL, LLLW, LLWL, LWLL, WLLL


The corresponding odds of each event happening are:


.50^4+4(.50^4)


= .3125


31.25% chance of losing three games in four contests played when the probabilities of winning and losing are equal.


With the combinations laid out on paper, the measure of streaks and cumulative probability can be manipulated to betray whatever event that one finds necessary to calculate the odds of happening.


To revisit the delicacy of a sequence of wagers, imagine a 10 game sample where we wish to find the odds of having AT LEAST one three game losing streak. Again the nature of how the streaks are formulated can be realized when typed. The set of runs we are evaluating play out like this:


games 1-3, 2-4, 3-5, 4-6, 5-7, 6-8, 7-9, 8-10


Over an occurrence of ten games, the sample size is not just ten games but it is relative to the run being calculated. In this instance, a streak of three games reveals a sample size of eight over the course of ten contests. Using that logic.


Given k is number of games, n is the sample size, r is the streak size, p is the probability of r resulting in success, and X as success or failure:


k = 10
r = 3
n = k – (r - 1) = 8
p = .125
X = count of success (1) or failure (0)


The variables now being defined, and a two outcome event of success/failure being a binomial distribution with a change in the sample size upon the observation of one possible event (without replacement), then to find the cumulative probability (any time r is observed as success), one can find the solution by using a binomial distribution function calculator, and the resultant number, being cumulative, is expressed at any point:


p(X > 1)


And by way of calculator, inputting the variables above, the answer:


p(X = 1) = 39.27%
p(X > 1) = 65.64%


This says there is roughly a 65% chance of losing three straight over the course of ten games under an equiprobable, two outcome environment.
Here are the critera for a binomial experiment, verbatim from the binomial calculator website link from above:

• The experiment consists of n repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by P, is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

We will assume the two probabilities here are independent outcomes, implying games follow a Markov Chain, which to simplify, basically means there is no such thing as momentum, and probability of two variables do not change over the course of immediate time.


In order to calculate only the odds of one three game winning streak happening, the formula is rather direct, and is referred to as the binomial probability rather than the cumulative probability (C is the indice of Combination and is a value every possible way a three game losing streak can run through ten games, either write them down on paper or use excel/calculator):


b(X; n, p) = xCn * px * (p-1)n-x

From above its stated the answer when p(X = 1) equates to 39.27%, and the intermediate math of the above equation plugging in the given variables is in agreement with what was shown.


The cumulative binomial probability is the sum of all binomial probabilities of success.


∑ xiCn * pxi * (p-1)n-xi

The eminent point here is delicacy of a wager. Even one with equal probability, the likelihood that over ten games a three game losing streak occurs at least once is about two thirds.


Now in writing this, I’ve realized the issue with the sample size as it integrates with runs. Since runs can overlap (i.e. a nine game winning streak is not three seperate three game winning streaks, but rather seven different ones), then the numbers probably aren’t as accurate then what is shown here. And I guess what one would do is to removing each possible winning streak over the course of the sample size that is greater than three, and isolate only the three game winning streaks, so maybe:


∑ xCn * px- [pr+1 + pr+2 ... pn] - (p-1)n-x

This would certainly lessen the binomial probability and cumulative binomial probability. It seems reasonable enough, inherently though it appears overly complex, there is hopefully an easier method.


Taking the 12.5% odds of winning three games in a row and appropriating that number over 100 games, which means a sample of 98 possible observed events, then that 12.5% equates to 12.25 total streaks, which would account for 36.75 losses out of the 50 expected total losses at even probability. This appears to be somewhat ridiculous. Three game losing streaks accounting for 73.5% of overall losses? I don’t know maybe the binomial distribution function is immune to overlaps.


Oh well, I’m sure somebody knows or has arrived at this issue before. Since I had already taken the time to write all of this I figured I would just keep it up here and address the problem and hoping to illicit some sort of response that leads to a desired resolve.


Ultimately, to diminish the emotional impact of seeing wins and losses transpire over the course of a bet slip, ideally one would create a +EV model, then develop a complex programming script that automatically extracts the best line from a variety of sportsbook and places the wager accordingly with the game that the model spits out. Then the overall ROI can be checked perhaps once or twice a month.

[uva3021]

Subscripts and superscripts were not formatted when I transferred over from html, so here are the equations:

∑ xCn * p^x- [p^(r+1) + p^(r+2) ... p^n] - (p-1)^(n-x)


∑ xiCn * p^xi * (p-1)^(n-xi)

b(X; n, p) = xCn * p^x * (p-1)^(n-x)

[d107]

this looks like a lot of work. how do you think this translates in to winning compared to traditional handicapping methods ??

[IrishTim]

Originally Posted by d107

this looks like a lot of work. how do you think this translates in to winning compared to traditional handicapping methods ??

Absolutely without a doubt. In my opinion, the age of the "seat of the pants" handicapping where you do your work by watching games and "feeling" out which teams will win/cover is over. I don't know of any long-term winners/pros who don't use at least some fundamental mathematics.

[IrishTim]

I too wish other people would comment on your posts. I think they're excellent and enjoy reading them.

Have you ever charted your streaks? I really believe that events in sports are correlated somehow, but I can't even begin to explain why. But if you go back and look at your records, I bet you'll find more 9-0, 11-1, 2-15,0-14 type streaks than a Gaussian distribution would predict for your sample. My experience at least.

[uva3021]

well over the long haul certainly one would expect records to be gaussian (central limit theorem), but short term if you take a collection of gamblers than not sure, of course there will be some variance in measuring sample to sample

It would be interesting if over the short term tracking a handicapper's record was more likely to be asymmetric rather than a normal distribution

just to put in perspective, here are the descriptive statistics of my bets from 12/09 - 04/10, using only games with a line of 140 to -140, and assuming all bets are flat wagers, in order to simplify

There were 200 such wagers over that 4 month stretch (less pushes), again to simplify, let's say the expected W/L record is 100-100, therefore the expected mean is 100, and my actual was 105 - 95



What the last row says basically is that there is roughly a 12% chance of somebody winning at least 105 wagers or more, and this number approaches 50% closer to 100, so the data is hyper sensitive, and obviously I didn't account for the varying probabilities across the magnitude of 140 to -140, but I figured it would be easy to see the possible expected distribution, after making the somewhat irrational and arbitrary assumption that the +140 and -140 wagers cancel out

Because of the small sample size there will always be issues with the data

Now here is a straight graph of all my wagers, 236, assigning a Win as 1, push as 0, and a Loss as -1

The black line synchronized with the graph is the 20 bet moving average, and the straight line moving slightly upwards is obviously the trend line.



My worst stretch over a 10 game minimum by percentage was 2-9, but the graph demonstrates I fluctuated between states of suck and awesome until ultimately being rather serviceable

[IrishTim]

I like how you have that done UVA. I should graph my records like that. What program did you use?

But only 235 wagers in 4 months? Methinks you need to get that up. I avg around 300 a month.

[uva3021]

I just used excel

lol 300 a month

well it was around basketball season, not as many plays as baseball season where sometimes you feel like you are making 300 wagers a day

[Kramer]

Good stuff

LETS SAY YOU MAKE 21 BETS
WITH A 57% WINNING AVG
EACH BET
WHAT HAPPENS?


HOW MANY WINNERS COULD YOU EXPECT IN YOUR NEXT 21 BETS?
57% OF 21 IS 12 SO 12-9 RECORD .
Trouble is, a record of 12-9 is not the only thing that can happen.

There are several other results that are also very likely,

TROUBLE IS MANY OTHER THINGS CAN HAPPEN OF WHICH
SOME NOT SO GOOD

3-18 or worse 0.01% of the time
..4-17 or worse 0.04% of the time
...5-16 or worse 0.18% of the time
...6-15 or worse 0.71% of the time
...7-14 or worse 2.22% of the time
...8-13 or worse 5.81% of the time
...9-12 or worse 12.82% of the time
...10-11 or worse 24.21% of the time
...11-10 or worse 39.61% of the time
...12-9 or worse 56.98% of the time
...13-8 or worse 73.25% of the time
...14-7 or worse 85.83% of the time
...15-6 or worse 93.77% of the time
...16-5 or worse 97.80% of the time
...17-4 or worse 99.40% of the time
...18-3 or worse 99.88% of the time
...19-2 or worse 99.99% of the time


It is just as easy to go 7-14 2.2 % of time
as 16-5 ...

It is much more difficult to get off your feet after going 7-14
and keep doing whast you'/re doing with the same expected
12-9 result ...

[Grizzly]

Originally Posted by IrishTim

Absolutely without a doubt. In my opinion, the age of the "seat of the pants" handicapping where you do your work by watching games and "feeling" out which teams will win/cover is over. I don't know of any long-term winners/pros who don't use at least some fundamental mathematics.

Agree and disagree here.

To preface I'm talking purely NFL here and I do use mathematical concepts and yes they have raised my winning percentage by a point or two over the last five years.

I agree that using mathematics, probabilities, using hold percentages, streaks, and several other mathematical handicapping concepts are of extreme value and definitely have their place in the sports markets but to me they are the parameters not the meat and potatoes of winning and losing. Purely math can by itself win without a doubt but the books have these concepts and employ them with much more accuracy, better technology, and with many better minds than you or I.

Ineffiencies can definitely be found with math but understanding the key concepts of why teams win and lose, the emotional elements, football schemes and their strengths and weaknesses, the value of certain positions compared to others, and the ability to see how these concepts translate onto the field and quantify them into a number along with the understanding of key numbers are, to me, much more important for raising your winning percentage and profits.

Now this isn't saying that it can be done just by watching a few games, reading some magazines, watching Fox pre-games and looking at boxscores. The vast majority of bettors lack the knowledge and experience to be able to translate a teams scheme, personnel, and critical match-ups inside of those schemes against another opponent.

Been at this for nearly twenty years and spent the first 8 or 9 placing $10 to $20 wagers and playing "Pigskins" (parlay cards) with the locals trying to figure out the how's and why's. Kept notebook upon notebook of stats, notes from games watched, and observations of the critical elements of why a team won or lost. Created thousands of sets of power ratings over those years tracking them and witnessing varying amounts of success and failure.

At that time it wasn't necessarily to make money but for bragging rights amongst football crazed friends and family members who also wagered. The competitiveness of that atmosphere and thousands upon thousands of discussions dissecting games and how to win led to a crystallization of how to beat the bookie way before I used any in depth mathematics. And to this day just using the fundamentals I learned over those many years will outperform any purely mathematical formula or combination of mathematical concepts I have found.

I would recommend just math to someone just starting out because it is a safer, faster, and more predictable outcome but studying the game and it's nuances is still the key to rising over the 53% plateau and not having to place 50+ wagers a week especially in the NFL where market inefficiencies are much harder to come by.

Enjoy what you guys do here and reading the vast amount of useful information so not trying to dismiss it in any way and I agree it is another tool that should be used by anyone serious about winning.

Just my two cents.

[uva3021]

great post Grizzly, it appears you've been afforded the "luxury", i guess you could call it now that its over, of taking the opportunity to struggle through the learning process and the humility that comes with it.

such situations typically cultivate character and stabilize emotions, and foster the visceral nature of how you can evaluate performance, i would assume about mid way through the season you can almost to some degree anticipate the nature of the team with all the consideration of a coaching staff

that's the advantage of keeping notes and staying aware of the environment for which you have applied yourself

[Grizzly]

Thanks Uva. Enjoy all your material.

Actually experience has taught me that the key to the NFL is the early season before the books adjust and while public perception is still based on the prior year. To me it's a year long passion and finding that '09 Denver or Tennessee or Pittsburgh team that the public, the media, and the books either overrate or underrate is one of my obsessions. Find them and you will have at least three to four very good weeks to begin the campaign.

By midseason the lines are much tighter and it comes down to value, emotional elements, mathematics, teams who are over or underrated because of either very tough or very weak early schedules, injuries, and pure personnel match ups. By that time my power ratings also have enough information to begin producing a decent amount of winners.

And yes knowing the coaches and coordinators and their tendencies are critical but it begins from the time they come in the league not just at the start of each new season. I want to know how they are going to play before the season starts. It is a lot of research and watching games but with the internet and all the replays of games on the NFL Network things have become much easier.

Look forward to reading more of your posts and threads, always very informative.

[IrishTim]

Agreed, UVA is among the best posters on all the forums. I think your initial post in the thread was pretty accurate. Certainly information is paramount, but if you don't have your own projection of what the line should be, it's hard to know if you're getting value or not.