## Mathematics when making bets at bookmakers.

As you know, they start a game of chess with white pieces. White pieces make the first move, and therefore the initiative in the opening is in their hands. The same situation is observed at the online bookmakers. Shaping the odds on the upcoming event, the bookies are first to start the gambling. We also enter the game but with black pieces, and we are trying to beat the bookmaker by his own means.

In this article I will present my vision of the proper game with black pieces. I must say that the material below contains terms, many of which, I am sure, you do not know. Therefore, to avoid “information clutter” I split article by theses, each of which not only explains a separate term, but is a logical continuation of the previous thesis. So, let’s start.

**Margin** – guaranteed profit of bookmaker obtained by artificially reduced Odds for each of the possible outcome of the event. The fact that the probability of each outcome can be determined mathematically, and if the probability of all mutually exclusive outcomes is represented in fractions of a unit, then we obtain a number of X, which when substituted in a ratio of 1/X give us the required ratios.

To understand what is at stake, I will give an example of a popular coin-tossing. As you know, the coin has two sides. The probability that the coin will fall, “eagle” up, as well as the “tails” is of 50% or 0.5. Hence, the odds for each of two possible outcomes is 1 / 0.5 = 2. So, if bookmakers offered us to guess how the coin will fall, providing an opportunity to bid by a factor of 2 for countless number of times, it would not earn them a penny. And as soon as both Odds 2 are reduced at least to 1.99, the bookies are even if in scanty, but the payoff. We, in turn, are the losers – of course, when it comes to the multiple repetition bets or to put it otherwise, about the game over a long period of time.

Equilibrium Betting – bet on the outcome of sporting events with the odds that evens margin. The fact that the odds offered by bookmaker reflect the probability of outcomes. Often, the objective analysis of the bookmakers is quite good, and we have no reason not to trust them. Yes, they also can be mistaken, but it happens so rarely that we’d better not go on about our intuition. However, sporting events, betting on which is mathematically justified, exist! Most often are the situations with Odds “born” in the competition. If you do not understand, I explain literally: popular beginner mistakes of bookmakers are overpriced bets for one of the outcomes meant to entice players into their own diocese. Naturally, the odds on the reverse outcome is understated to keep the margin, but the balance is already broken – as soon as the odds the fact that the coin will fall with “eagle” (“tails”) up becomes 2 or more, we are betting without hesitation. And even if in a particular case we lose, believe me, in the long run we will win.

The second part of the event of an equilibrium betting is the situation with the adjusted factors – since their formation and before the sporting event starts. Those of you who are familiar with bookmakers, you know that the Odds are in constant dynamics, that is a subject to change at any time. These changes have two types: wave (increase – decrease – increase and vice versa) and constant (increase – increase – increase and vice versa). The majority of players are profoundly mistaken, seeing the cause of such changes in an attempt of bookie to prevent them from winning easy money. “The bookies know something, so they modify ratios” – they naively believe. In fact, everything is much easier.

- First, if the bookies learn about any arrangements between the participants of the upcoming sports events, they either drastically reduce the maximum amount of bookmaker bets, or remove the bid altogether.

- Secondly, remember once and for all: bookmakers absolutely do not care how a sporting event ends – their interest is in guaranteed profit margin over a long period!

And here we have a “loophole”. The fact is that the effect of the margin becomes the maximum in the case when the balance of bets on mutually excluded outcomes equals to none. In other words, if ten people did Betting on the coin falling with “eagle” up, and the ten men did Betting that the coin will fall “tails” up, and the Odds in this case are equal to 1.9, then with equal stakes on both sides bookmakers get the optimum effect. You may notice that the most advantageous situation would occur if all twenty people have entrusted their fate to the “eagle” and the coin fell to “tails”, and I agree with you, but with one caveat: this is an isolated incident, and bookmakers cannot rely on it.

Now back to the dynamics of factors, why it is happening? To make it clear, let us imagine the scale, one of which is the sum of the bets of bowls that the coin will fall, “eagle”, and on the other – the amount of betting on the outcome of with mutually exclusive results. As I said, for the bookmakers the optimal balance is betting on mutually exclusive outcomes with zero, and they are trying to fully support it, that is to balance the scales. The bookmaker has already made his move, giving us the opportunity to bid on a particular outcome, and now he must wait, as we were going to lead. At some point, one of the bowls starts to outweigh the other so that the bookmaker decides to intervene. Since almost all the players give the advantage to the same end, the bookmaker has to make adjustments for the following reason: in the projected outcome there is a strong imbalance, and the risk that the majority guess the outcome of events dramatically increases. In order to “balance the scales,” bookie changes the Odds, increasing them to “unpopular” outcome. At this point, we have to determine for ourselves whether the adjusted odds conforms to equilibrium betting, and if so – to make a bet.

Here I digress to briefly justify this action. Bookmaker determines the probabilities of mutually exclusive outcomes and genebets Odds. Bookmaker artificially lowers bets for margin – his guaranteed profits over a long period. At some point, he was forced to increase the one of the Odds (the probability accounted initially remain the same), and we get the equilibrium rate. Immediately I note that in this situation, there is one drawback – you need to know the initial Odds derived for the mutually exclusive outcomes of sporting events – to monitor the dynamics of their changes. Roughly speaking, if you missed the moment when the event appeared in the line and it took some time, the assessment of the dynamics will not be objective.

**Bankroll** – money capital, designed specifically for the game at the bookmakers. Once you finally decide that making betting bets, being a professional player, not an “idle onlooker” who loves to tickle his nerves, means you will have to clearly define the amount of money allocated for the game at the bookmakers. The fact that many beginners make the same mistake, shifting money “from one pocket to another”. Here’s how it goes: After a series of successful bets player starts to put “easy” money on everything, or vice versa, having suffered a series of setbacks, one stroke pulls the rest of your bankroll by betting an amount several times greater than the previous ones, and, not having achieved a positive result out beyond bankroll, continuing to play already on money intended for other purposes. In the first and the second example, the end looks rather baddie: Playing over period of time, you will at some point be left with nothing – that’s the price of managing your bankroll badly!

One turn bankroll – the number of betting bets that generate a total of bankroll. Depending on the game, one turn of bankroll is either constant or variable.

**Fixed rate** – model of the game, in which the amount of betting bets remains unchanged within a single turn of bankroll, which remains constant. For example, your bankroll is $ 100, and the amount of the bookmaker bets $ 2. One turn bankroll in this case is 50 bets. As soon as the latter will be calculated, the fiftieth position, your bankroll will change up or down. In proportion to the change the amount of the fixed rate will vary, so as to remind one turn of bankroll is constant. So if your bankroll will be $ 300 flat rate will increase to $ 6. Just note that you can derive profit from the game, or replenish your bankroll only on the boundary turns, when changing the amount of bankroll does not affect the ratio of the “bid amount – bankroll.”

**Unlimited Multiple bet** - model of the game, in which each successive Betting genebets conditional multiple bets along with the previous or the bets are placed simultaneously within a single Multiple bet. One turn of bankroll here is variable and the amount of the initial bet in each Multiple bet constant. How does it look? Your bankroll is $ 200. You do bookmaker bet of $ 2 on the outcome of a factor of 3, and it plays. The position is calculated, and you have now 204 dollars. The next bet on the outcome of a factor of 2.5 is already $ 6 (the amount received in the calculation of the last position). This time you lose. Bankroll is $ 198, and the subsequent rate again will require $ 2. Here I want to explain something.

- First of all, the word “Unlimited” does not denote the number of items or general factor, but the length of time when a chain of sporting events within a single Multiple bet lasts.

- As for the number of positions, it should be the same throughout a single turn of bankroll. Personally, I recommend form of multiple bets from three to four events. What does this mean? Let’s say you chose the model of the game “unlimited multiple bet made up of three positions.” In this case, you return to the initial bet in two situations: when the unsuccessful outcome occurs or a successful outcome at three in a row. Given that one turn of bankroll is a variable, you can withdraw the profit from the game or replenish your bankroll at any time, but under two conditions: in the absence of none calculated bets and in compliance with the ratio of the sum of the initial bet to the amount of your bankroll, for example, 1 to 100.

Of course, you have a question: which model is preferred and why? Take two bankrolls of $ 200 each. In the first case we are playing at a fixed rate, in the second model we choose “Unlimited multiple bet of three positions.” For the convenience of mathematical calculations we take the average rate on the equilibrium outcome of a factor of k=2.2 with the probability of winning 50%. For the first case the calculation is simple. Suppose we make 40 bets per $ 5. At the end of one turn of bankroll we have 20 positions, calculated with the result “to pay $ 11″ and 20 losing bets, and hence our bankroll grew and contains $ 220. Now let’s consider the situation with unlimited multiple bet of three positions. Given the fact that to win on a separate segment, we need to predict the outcome of three bets in a row, both the desired rate and the probability of a successful outcome radically change. Now the factor is k=2.2*2.2*2.2=10.65, and the probability of winning is 12.5%. In other words, we get a positive result in only one of the eight multiple bet bets. As before, we make 40 bets per $ 5. And only five Bets promise us the pot. However, at the time of settlement of all bets our bankroll will contain no less than $ 266.

As you can see, the model of “unlimited multiple bet of three positions” brought us a profit that exceeds the amount obtained as a result of the game at a fixed rate by three times. But do not jump to conclusions. The fact is that in the first case we needed to allocate of all sporting events 40 outcomes with positive expectation just to increase your bankroll by 10%, and in the second case we increased your bankroll by 33% just as the result of 120 equilibrium bookmaker bets! Not hard to guess that the outcome of the game at a fixed rate over a similar period will differ little. Therefore, from the viewpoint of mathematical analysis both models are similar. The main difference lies in the ability as a player to reach the line of sporting events and to find outcomes with a positive expectation – the more experienced privateer, the more is he inclined to the model of “unlimited Multiple bet”, which allows for wrapping bankroll several times faster. Conclusion: If you are starting a privateer, then play at a fixed rate – learn, first of all, how to qualitatively determine the outcomes based on a positive expectation and to tightly control your bankroll.