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The time has come for the concluding 101 related to BR. Our initial mission since our appearance in the Jungle has been to, in *simple* terms, present this risk-free way of building a bankroll/earning easy money for those of you who have not made it yet. At this point in time we are content with the fact that not much remains until all the necessary material to successfully complete the ‘scheme’ is available at the Substack/Website and it should not be long before all of your bonus rugging operations out there are up and running.

*If you have made it already (net worth > 1 M $ + at least two profitable businesses running), obviously a couple of thousand dollars will not change anything → feel free to do whatever you like with the information. Nevertheless, if you are one of the few who have actually made it you are probably also the kind of guy who will find the execution of the concept to be a fun pastime. *

If you are not interested in more BR-material, go to the ‘General information’-section at the end of the post to find a brief discussion of what is to come.

*Note: Giving Qualifying Bet 101 and Freebet 101 a read prior to proceeding with this text could prove to be beneficial, albeit not strictly necessary. *

### The risk-free bet

Let us avoid wasting any time and instead jump straight into the fundamental question of this piece, namely, what is a risk-free bet?

As the name suggests it is an offer provided by a bookmaker to place a bet *with no risk*. If you win you keep your winnings, if you lose the stake is returned to your account. No matter the outcome, you *do not lose* any money.

Assuming fair odds for the sake of simplicity, the expected value of such an offer is calculated by the below formula (explanation here), with W being the size of the risk-free bet.

*Need to refresh your understanding of EV and how to calculate it? Visit our Betting 101! *

Note that as the ODDS → ∞ the expected value of the risk-free bet converges to its upper bound, the nominal* value of the offer. Furthermore, an interesting observation is that this expected value is the exact same as the one in the freebet case which indicates that a freebet and a risk-free bet seem to share some similarities.

**Nominal value is a financial term for the ‘value’ of a security that is set by the company issuing it; unrelated to market value. Similarly the nominal value of a risk-free bet is the ‘value’ assigned to it by the betting company; again unrelated to its real world dollar worth. *

*Autist note: In fact, a true turbo would claim the two types, freebets and risk-free bets, are equivalent since expected values, payoff functions etc. are identical. The only thing that differs is a meaningless transfer of money back and forth between your bank account and your balance with the bookie. Think about it and you will notice that a risk-free bet may in fact be viewn as a freebet with the extra condition that you must first transfer W $ to the bookie to ‘qualify’ for the freebet, only to at a later stage withdraw the same value of W $ no matter what happens in the game, of course assuming you request a withdrawal immediately after your risk-free bet has been settled. Since everything is identical math-wise, you will notice that the formulas/manipulations/operations performed in this post’s theory-PDF are more or less ‘copy pasta’ from the one attached to the freebet-guide. *

### Hedging a risk-free bet

*Note: As stated in the autist note above, a risk-free bet is essentially identical to a freebet and to avoid wasting precious time on this section we have basically copied the material from the Freebet 101 and added some minor modifications. *

Suppose you are now in the following position. You have recently signed up with a bookmaker and the welcome offer provided by the bookie is a risk-free bet with a nominal value of W $. How do you make sure you extract as much of the expected value as possible from this proposition while simultaneously being outcome-independent (fully hedged, equivalent profit regardless of outcome)?

#### Case 1 - Betting exchanges available (Europe, Australia, …)

As usual the easier case, however not as immediate/obvious as for qualifying bets. The question we would like to answer is, given a certain number of risk-free bet dollars wagered on an outcome, how much should we lay on it to convert the volatile proposition into a risk-free one?

*Quick explainer:*

*Lay betting is essentially betting on something not to happen,*

*Lay odds – the odds which you are prepared to offer someone wanting to place an ordinary back bet.**Backer’s Stake – the amount you are prepared to let the backer bet with you. This is your potential winnings.**Exposure – (Lay Odds - 1) x (Backer’s Stake). This is the amount you are risking.*

With some mathematical modeling and a few clever manipulations we arrive at the below formula. For the curious ones (should be or NGMI) the underlying mathematical theory describing how to derive such a solution can be found here.

Additionally, by using the above result we may construct a graph describing the profit function under the assumption that you are taking advantage of a 1000 $ risk-free bet offer and use an exchange with a 2 % commission, i.e. values ordinarily used in practice. The result can be seen below.

*IMPORTANT: Note the necessity of choosing low probability/high odds events for your risk-free bets unless your mission is to waste money. Other things being equal it is always more favorable to aim at the higher odds ranges. However, at odds around 3.80 - 4.00 the curve begins to flatten out at a rapid pace and therefore we would say that the ‘risk-free bet floor’ lies somewhere in this region. *

A calculator implementing the above mathematics can be found here.

#### Case 2 - No betting exchanges available (USA, Canada)

Slightly more demanding. If no betting exchanges are available you will have to employ other bookmakers for bet hedging purposes. In our upcoming practical guide we will provide recommendations of free betting software to help you find which bookies are offering the top market odds in case you are not comfortable doing this manually, all to make sure you are not wasting precious dollars betting at worse odds than available elsewhere.

Either way, we state the formulas for the ‘no-exchange‘ case below and if you are interested in the math behind it you visit our PDF.

*Note: With some clever assumptions on the relationship between the odds on one outcome and the others we could come up with a similar plot/figure as in the exchange case. However, since the graphs would look more or less identical we abstain ourselves from doing more than repeating the fact that the strategy of *low probability/high odds events* is again optimal. *

A calculator implementing the above mathematics can be found here.

### Quick facts on risk-free bets

Similar to freebet offers, risk-free bets are very common in the US and lots of different bookies use it as a marketing tool to attract new customers.

Risk-free bets are on average worth ~ 70 % of the nominal value declared by the betting company, assuming an accurate strategy is applied.

A standard welcome offer in the US is a risk-free bet of W dollars, repaid in the form of a freebet if the bet loses. Since we have learned earlier that also freebets are worth ~ 70 % of their nominal value, some small adjustments to the formulas are needed to handle this composition of a freebet inside a risk-free bet. More on this in our upcoming practical guide!

### General information

To finish off today’s post, we have compiled a couple of things regarding both BR and betting in general.

**Bonus Rugging: **

Now that the three main BR 101’s have been made available to you, basically any offer out there should be fairly simple to understand and take advantage of. Much like any mathematical operation is built up by a combination of additions, subtractions and multiplications, a majority of all the possible betting offers one could imagine may be modeled as a union of qualifying, free- and risk-free bets. In the language of autists, one would say that the collection {QB, FB, RFB} yields a basis for the space of betting offers.

Remaining fundamental posts:

- Practical guide on BR

*(paid subscribers only*), expected within a few weeks. Will be heavy and full of details to make sure you optimize your BR-process and avoid making any unnecessary mistakes.**Update: This practical guide can now be found here.**- Explanation of the Master Calculator with surrounding theory

*(again paid subscribers only*). Will follow the same outline as the 101’s with the only difference being that it will require more advanced math knowledge/tools. Named ‘Master’ since it is constructed to take care of an arbitrary number of offers in any kind of composition, all at once, which in practice means that you could complete the whole concept within ~ 2-3 sessions.A Twitter Bot has been created to find and share appropriate games for BR on a daily basis. Visit @BowTiedBettorII! (

*update: main acc was banned → this one is used as new main with bot content daily at 10.15 EST*) More information on its relevance will be explained in the upcoming practical guide.

**Betting in general: **

As soon as all the required BR-material has been shared the stack will transition into more general betting territory. If you have a particular idea of a subject that could be of interest for both you and the rest of our readers, please share it with us in the comments/on Twitter.

Bayes will visit the BowTiedBettor Substack shortly.

To align ourselves with our subscribers we will expand our current betting operation and include at least one major American sport. Horse racing in Europe has been our primary market for many years and therefore it will be a refreshing challenge to see what we might achieve in markets we are currently not comfortable betting. As this is not mainly for our own purposes but rather for our readers’, we believe that you should be part of the decision process regarding which sport/market to choose. Please drop a vote in the poll below!

That is all for today!

Until next time…

**Disclaimer:** None of this is to be deemed legal or financial advice of any kind. These are ***opinions* **written by an anonymous group of mathematicians who moved into betting.