Casino Bonuses - Calculating Expected Value (EV)

Expected Value (EV) is a fundamental concept for a professional gambler - by calculating the EV of a bonus it allows you to understand in a rational and logical manner whether or not the bonus is worth taking on and whether over the long run you will be profitable by taking that kind of bonus.

In this article we will look at what EV is and how to calculate EV for a given bonus.


Expected Value (EV) - as it relates to gambling - is the long term average value that can be expected from a given bet. However, the term 'EV' is often misused when talking in Advantage Play terms - for example when talking about the 'EV' of a casino bonus, people usually take EV to mean 'the amount of the bonus that can be expected to be realized (ie withdrawn) after completing a wagering requirement'.

For the purpose of this article we will take EV to mean just that: the amount or value of a bonus that can be expected to be 'taken away' once the wagering requirement has been completed.

This article looks at calculating the EV of casino bonuses, with some discussion of other factors that affect EV.


How to calculate the EV of a bonus?
Calculating the EV of a bonus relies on knowing a few variables:

• B=how much the bonus is
• W=how much wagering you have to do (the 'wagering requirement' / WR)
• A=the house advantage of the game you're wagering on

From these variables you can very simply/crudely calculate the EV (E) as follows:

All you're doing is subtracting the amount the house/casino will take over the wagering requirement (the house advantage) from the amount of the bonus. This is a very rough rule of thumb - there are other factors that affect EV such as the variance of the game that you choose to complete the wagering on, the strategy you employ and the price you put on your time.

The most important thing to remember is to calculate the EV value of any bonus promotion before even considering depositing for it. Make sure that above all the EV is positive - you will often see people refer to a bonus as being +EV - by that they mean that the expected value of the bonus is positive, a profit would be made by completing the bonus wagering requirement all things being equal.

An example of calculating EV
As an example - a site has a bonus promotion offering a £30 bonus with a 100xWR and Blackjack can be used to complete the WR. This gives us:

• Wagering Requirement, W=£3000 (30x100)
• House Advantage, A=0.005 (very roughly assuming 0.5% for Blackjack, ie for every £100 you spend on BJ, on average the house takes 50p)

From this you can calculate:
So over £3000 of wagering on Blackjack, all things being equal you can expect to lose £3000*0.005 = £15. This is the 'house take' or 'house advantage' - if you were not playing with a bonus you would simply lose £15 over £3000 of wagering. However because you are starting with a £30 bonus, after £3000 of wagering you will be left with £30 - £15 = £15. So the expected value of the bonus is £15.

Expected value is the 'long run' average value you can expect from a given bonus - ie if you could play the same wagering requirement on the same game over and over an infinite number of times, the expected value is the average value you can expect to get back over all of those wagering sessions.

How variance affects Expected Value
The type of game that you complete your wagering on and the way in which you play those games have a very large part to play in the actual value of a bonus realized compared to the expected value. Some games/strategies will stay very close to the expected return whereas others can vary wildly in their return. The degree to which a game's actual value (ie the amount you can expect to win from each single bet unit) varies from it's expected value is known as 'variance'.

With a 'low variance' game (Blackjack for example), you can roughly expect the actual value of the bonus you extract to be relatively close to the 'expected value' of the bonus. However with a higher variance game (like a slot machine for example), the chance of busting out before you complete the wagering requirement is a lot higher.

Variance is an important concept to understand because it can dramatically affect the amount of value that can be extracted from a bonus. By completing the wagering requirement for a bonus on the lowest possible variance game, you increase the chance that you will extract somewhere close to the expected value of the bonus you are wagering for.

How strategy affects Expected Value
The way in which you play a game also has a large effect on the expected value - if you play in a way that optimizes your chance of winning for any given 'position' you find yourself in (ie your starting 'hole cards' in a game of Blackjack compared to the dealer's hole cards and the cards that are dealt to you), you limit the amount of variance for the game you are playing. As a result the strategy you employ when playing a game has an effect on the expected value of a bonus.

Many casino games have what is commonly known as a 'perfect strategy'. Put rather crudely - the 'perfect strategy' is one which will yield an optimal return on a bet, all things being equal. Importantly the term 'perfect strategy' should NOT be seen as some silver bullet 'system' that will 'beat the casino' - no such thing exists! - rather it should be seen as a strategy for minimizing the losses made on a casino game such that over a period of wagering the expected value can be achieved for that game.

By using the 'perfect strategy' on the lowest variance game possible, you increase the likelihood of extracting the full expected value from a bonus.

How Stake Size affects Expected Value
The size of bets you place on a given game also have an effect on variance and so in turn affects EV. By placing the smallest possible stakes on a game, all things being equal you minimize variance and as a result maximize the chance of seeing the EV be realized.

By placing larger bets, any variance inherent in the game/strategy you are playing is effectively amplified and as a result the overall variance is increased. To keep to the EV, use the smallest possible stake size.

How Stake Size affects Expected Value per hour
Time is a valuable commodity! In the previous section we said that sticking to the smallest possible stake sizes increases your chance of extracting the exact EV from a bonus. HOWEVER this was more of an academic statement than anything else - yes it is a fact that sticking to a minimum staking size will reduce the variance encountered, but it will also maximize the amount of time it takes to complete the wagering for a bonus!

By staking 10p/hand on Blackjack for a WR of £3000 at an average of 10 hands per minute - that will take 50 hours to complete the WR! If we use the example from above (EV of £15 for a WR of £3000 on Blackjack), that equates to an average profit of only 30p per hour, clearly not worth the effort.

As a result it is always important to consider how long it will take to complete a wagering requirement based on a given staking size and take that into account when calculating the return per hour (and at the same time remember that increasing the stake size will increase the variance experienced).

Summary
We saw above that the EV of a bonus can be roughly calculated given a few variables. This is the very first thing that should be calculated when considering completing a bonus wagering requirement. Check that the bonus has +EV - the expected value of it is positive and you will see a profit all things being equal.

We then went on to look at variance, how it affects the EV of a bonus and how choosing the correct games/strategies can affect the EV. Always try and pick the lowest variance games allowed and stick to the 'perfect strategy' for that game to optimize your returns.

Finally we looked at staking sizes and staking sizes have on return per hour. Whilst placing minimum stake sizes minimizes variance, it also maximizes the time taken to complete a wagering requirement. Finding a happy balance between stake size and acceptable variance is important in optimizing your return per hour.

TODO: add links to the various calculators around (ie beatingbonuses etc) (or maybe even have a separate page for all the various online calculators/tools?)