# How to calculate the EV of a sequence of bonuses.

#### 3Dice

##### I-Gaming Industry Representative
A lot of misinformation is available on this subject - partly because it is not as trivial as it may seem at first glance, partly because there is no clear definition of what we are trying to calculate.

So lets first try to define what we are exactly trying to calculate. The formula you'll find all over the web is not 'mathematically wrong' .. to be precise, it just doesn't calculate the number you need. To avoid confusion I wont repeat that formula here, but what it calculates is the EV of a bonus, _if you only take just one bonus your entire life_. If that is the masterplan, take a bonus once and then never again, then you can use the formula you'll find on many websites. (that formula and the strategy that goes with it does not maximize the EV, it minimizes the risk of ruin by 'grinding' which is playing low bet low variance.).

If however, you are a player that values the entertainment value of gambling, and are wondering whether or not on the long term you should or rather should not take bonuses, then you need an entirely different formula.

Unfortunately, the formula to mathematically determine, ahead of time what the EV of a sequence of bonuses is is complex, and moreover it requires information (like e.g. the numerical variance of the game) that a player typically does not have access to.

That doesn't mean you cannot calculate the EV of a sequence of bonuses. In fact, its really easy to do on past data. From a player point of view, you can use a simple algorithm that will update the EV of every next bonus taking into account all the past bonuses you have already received. Allow me to skip the explenation for a second and get right down to the stuff you need.

Algorithm to calculate the running EV of a sequence of bonuses.

Code:
`````` - Log all your sessions. For each session note :

- B = bonus.
S = stake.

- after each session, calculate the following numbers.
TB = total of all bonus.
TS = total stake over all sessions.

- after each session you can calculate the EV of the sequence up
to that point. (assuming a 5% houseedge here).

TOTAL_RUNNING_EV = TB - TS*0.05``````

An example ..

So lets look at a sample. We'll simplify things and have a player lose 9 times in a row, then win 1 time. He'll deposit \$50 every time, and get a \$50 bonus on top of it. He spins a reel of fortune that has 10 slices. 9 win nothing and one wins 9.5 times betsize.

Code:
`````` deposit 1 : 50D + 50B. total stake = 100.

The EV after deposit 1, using formula above : 50 - 5 = 45

deposit 2 : 50D + 50B. total stake = 100.

The EV now is 90

deposit 3,4,5,6,7,8,9 : 50D + 50B. total stake = 100 each time.

The EV now is 9*50 - 9*5 = 405

deposit 10 : 50D + 50B. total stake = 100, total win = 950

The EV now is 10*50 - 10*5 = 450

The player cashes out at this point. He deposited 500, cashes
out 950 .. exactly the EV of the bonus over his own deposit.

The total stake on the machine is now 1000, total win 950 .. i.e. 95% machine.``````

Also notice that if our player had decided to not take a bonus on that last deposit, but just play his own 100, he would still have an EV of 400 from past bonuses ...

Also notice what a max-cashout would do here .. a 50 bonus with 10x max cash sound reasonable ? .. in this case it would COMPLETELY eliminate the player advantage ..

In fact, in this scenario a maxcash of 10x (or 500) would be the equivalent of a WR of more than 200x. (stay away from maxcash bonuses!!)

The rule of thumb.
Code:
``````[COLOR="Red"][SIZE="4"] [CENTER]For as long as 5% of your totalstake is lower than your
total bonus, you  are playing at positive EV.

[/CENTER]
[/SIZE][/COLOR]``````

Playing without bonus in a sequence of bonuses.

What is the effect of making deposits where you don't claim bonuses inbetween those where you do take bonuses ? Well, the formula stays the same .. total bonus will not increase, but total stake will. In the example above, if our player had not taken 10 bonuses, but only 5, alternating a deposit with a bonus and a deposit without a bonus, then at the tenth session he would have an EV of :

EV = 5*50 bonus - 1000*0.05 = 200

And so when he cashes out on the tenth session (after 5 deposits of 50 on which he claimed bonuses and 5 of 100 without bonus), he cashes out 950 on a total deposit of 750 .. or again exactly the EV ahead when the machine is at 95% (so he didn't get lucky - he got the exact average RTP).

In other words depositing without a bonus doesn't instantly mean you have no EV from past bonuses anymore. As long as you follow the rule of thumb above - you are playing at a positive EV.

Closing thought.

It's lady luck that gives out the best bonuses. It's not the best mathematicians that win the most - its the luckiest players.

Cheers,

Enzo

#### Wibbler

##### Dormant account
Um, if you're playing without a bonus then you're just gambling. And your example is just plain ludicrous - you pulled the "lose 9 then win 1" sequence out of thin air, and I've only heard of a handful of bonuses that allow you to cash out after just wagering 2xB.

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

#### liquuid_fusion

##### Meister Member
PABaccred
PABnorogue
PABnononaccred2
My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

Of course playing without a bonus is -EV, but what he's saying is that playing without a bonus does not necessarily cancel out the times when you did play with a bonus, ie over all your past play (bonus and non-bonus) your advantage may still be positive.

#### Jufo

##### Three-toed sloth
Of course playing without a bonus is -EV, but what he's saying is that playing without a bonus does not necessarily cancel out the times when you did play with a bonus, ie over all your past play (bonus and non-bonus) your advantage may still be positive.

Of course the advantage can still be positive as long as the total -EV from non-bonus play is in total smaller than the +EV from all bonus play but why play without bonuses if the EV maximized by only playing with bonuses?

Also the EV doesn't apply to past results. EV means "Expected value" of a result that has not yet happened.

An example: You flip a fair coin ten times. Before you start flipping, the expected number of heads is 5. Suppose that after 9 flips the coin has been tails 9 times. Before the 10th flip the expectation of heads is not 5 anymore but 0.5.

Similarily if you repeat a bonus ten times (\$50 D, \$50 B) with EV of \$45 per one bonus, then the total EV before starting playing is \$450. But if you lost your deposit first 9 times then at this point the expectation is not \$450 anymore but instead it is -405\$.

#### 3Dice

##### I-Gaming Industry Representative
Um, if you're playing without a bonus then you're just gambling.

Not quite .. explain to me what the difference is between :

1. deposit \$100, get 100% bonus, then deposit another \$100

2. deposit \$200, get 50% bonus.

By making the followup deposit, you effectively transform the 100% bonus on the previous deposit to a 50% bonus on both deposits. Again, to know if you still have EV use the rule of thumb printed above.

And your example is just plain ludicrous - you pulled the "lose 9 then win 1" sequence out of thin air, and I've only heard of a handful of bonuses that allow you to cash out after just wagering 2xB.

Would you care to give a reason why this is a "ludicrous" example ? It is just a simplified version of a slot if you like, at a 95% RTP. And the wagering requirement is irrelevant. You don't have to cashout on a deposit that came with a bonus even to take advantage of the EV.

The only given is that on the long term the machine pays out 95% of what was put in. So the casino makes 5% of the total stake. If 5% of your total stake is less than what the casino gave you in total bonus, then you have a positive EV. You will continue to have that EV on subsequent bonus-free deposits until such point in time where 5% of your totalstake becomes bigger than your total bonus.

My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

The point of my post is exactly to eliminate this wrong train of thought. Perhaps you should work out some number samples for yourself to see where you are wrong.

Kindest Regards,

Enzo

#### 3Dice

##### I-Gaming Industry Representative
Similarily if you repeat a bonus ten times (\$50 D, \$50 B) with EV of \$45 per one bonus, then the total EV before starting playing is \$450. But if you lost your deposit first 9 times then at this point the expectation is not \$450 anymore but instead it is -405\$.

That's not correct Jufo. You should've heard some alarm bells when you say -405\$ EV on a \$100 deposit . I could've understood you (wrongly) thinking the EV at that point is just \$45, but -405\$ ?

You can't just ignore the fact that most of the money you lost up to that point was bonus .. that's why I demonstrated the reverse effect. If we repeat the scenario indefinitely, the player will cashout \$950 for each \$500 he deposits. Hence the EV is \$450.

TotalWithdraw - TotalDeposit = TotalBonus - Totalstake * 0.05

This relationship needs to be maintained for the machine to be a 95% machine.

Kindest regards,

Enzo

#### Jufo

##### Three-toed sloth
Not quite .. explain to me what the difference is between :

1. deposit \$100, get 100% bonus, then deposit another \$100

2. deposit \$200, get 50% bonus.

If no play takes place between the deposits in option 1, then options 1 and 2 are mathemtically equal. However the option 1 without the follow-up deposit is better for the player than option 2 so the only effect that the follow-up deposit has, is to decrease the bonus value, in other words it is -EV to make such a follow-up deposit.

By making the followup deposit, you effectively transform the 100% bonus on the previous deposit to a 50% bonus on both deposits. Again, to know if you still have EV use the rule of thumb printed above.

Yes, player still has +EV after the follow-up deposit but less +EV than without making it.

You will continue to have that EV on subsequent bonus-free deposits until such point in time where 5% of your totalstake becomes bigger than your total bonus.

Theoretical EV on past outcomes has no real value to the player. The player either finished below or above EV and made a gain or loss. The result is already known and there is no expectation left in past results.

3Dice said:
Wibbler said:
My rule of thumb - you're only playing at +EV if you're playing with a bonus. Any other form of play (unless you're an expert card-counter or poker player) will on average just be handing money into the casino's pockets.

The point of my post is exactly to eliminate this wrong train of thought. Perhaps you should work out some number samples for yourself to see where you are wrong.

Could you clarify what exactly your point is? Are you saying that the player has +EV compared to his present situation by depositing and playing without bonus? And if not then why is Wibbler's train thought wrong?

#### Jufo

##### Three-toed sloth
That's not correct Jufo. You should've heard some alarm bells when you say -405\$ EV on a \$100 deposit . I could've understood you (wrongly) thinking the EV at that point is just \$45, but -405\$

Nope, it is correct. The player has deposited \$50 nine times and lost all of the deposits so he is standing at -450\$. The expectation of the 10th bonus is +45\$, making his expectation at this point -405\$. See the coin flipping example if you didn't get it.

You can't just ignore the fact that most of the money you lost up to that point was bonus .. that's why I demonstrated the reverse effect. If we repeat the scenario indefinitely, the player will cashout \$950 for each \$500 he deposits. Hence the EV is \$450.

The EV is \$450 only before any playing takes place, not after the fact.

This relationship needs to be maintained for the machine to be a 95% machine.

The player might need to play infinitely many times to reach the expected profit of \$45 per bonus, and never without a bonus.

#### 3Dice

##### I-Gaming Industry Representative
If no play takes place between the deposits in option 1, then options 1 and 2 are mathemtically equal. However the option 1 without the follow-up deposit is better for the player than option 2 so the only effect that the follow-up deposit has, is to decrease the bonus value, in other words it is -EV to make such a follow-up deposit.

It decreases the value of that bonus - obviously - but its still +EV towards not having taken a bonus at all.

Theoretical EV on past outcomes has no real value to the player. The player either finished below or above EV and made a gain or loss. The result is already known and there is no expectation left in past results.

Surely you can't be serious. There is +EV until houseedge*totalstake > bonus given. The time paused between spins (wheter it be minutes or weeks) is absolutely irrelevant.

Could you clarify what exactly your point is? Are you saying that the player has +EV compared to his present situation by depositing and playing without bonus? And if not then why is Wibbler's train thought wrong?

I'm saying the player has +EV compared to the situation of taking no bonus at all.

Surely if all you care for is EV then you should probably only take signup bonuses, never deposit without bonus, switch casino's more often than underpants. In doing so you'll have a load of installing/uninsatalling/administration - not to mention heartache and stress trying to cashout from the shady places this strategy undoubtfully takes you.

If however you care for other parameters like smooth cashouts, good service, no hidden or difficult T&C, quality of entertainment, then you will find that the occasional inbetween deposit without bonus is a requirement, and as demonstrated doesnt mean you can't still be playing at a positive EV.

This is not written for those who would like to abuse these bonuses and are playing only for that advantage. This is written for those people who value gambling as entertainment and to give them the reference frame they need to judge bonuses.

Kindest regards,

Enzo

#### 3Dice

##### I-Gaming Industry Representative
Nope, it is correct. The player has deposited \$50 nine times and lost all of the deposits so he is standing at -450\$. The expectation of the 10th bonus is +45\$, making his expectation at this point -405\$. See the coin flipping example if you didn't get it.

Jufo, your machine doesn't end up at 95%. The coin flipping example does not apply. I'm not trying to predict a future outcome. I'm just saying on average player wins 1 in 10 sessions. Consider it all in the past if you like. I'm just looking at the full sample. A new outcome adds to the full sample. The full sample will end up at 95% payout. When it does, the player will have withdrawn \$450 of the total of \$500 in bonus given. Anything you suggest that doesn't end at a total EV of +\$450 for a total stake of \$1000 half funded with bonus - is not correct.

again .. as long as totalstake * 0.05 < bonus .. you are at +EV and with neutral luck (ie machine at 95%) will still end up cashing out.

Cheers,

Enzo

#### just play

##### closed account
Enzo, can you "dumb down" your first post for people like me that don't understand?

#### Wibbler

##### Dormant account
I'm saying the player has +EV compared to the situation of taking no bonus at all.
So you accept, then, that playing without a bonus is -EV

#### blankley

##### Dormant account
Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

#### 3Dice

##### I-Gaming Industry Representative
Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

That depends - are you in it for just the EV or do you also consider bonus a means to prolongue your play and thus increase the entertainment value ?

Looking at it from that point of view one could state that the only way to lose the EV of a bonus is by replacing it with entertainment.

Kindest regards,

Enzo.

#### blankley

##### Dormant account
I believe that bonuses are for entertainment value only and people should never cash out after having been credited with one as that would be wrong.

I also believe the children are our are future, teach them well and let them lead the way.

#### KariRunk

##### Dormant account
I don't see the point of this thread. Since there's almost always wagering requirements for bonus, you need to take that to account when counting EV. Which makes almost all bonuses -EV for player and +EV for the casino.

And if the slots really are truly random, the past results doesn't have anything to do with the future EV.

#### BingoT

##### Nurses love to give shots
I don't see the point of this thread. Since there's almost always wagering requirements for bonus, you need to take that to account when counting EV. Which makes almost all bonuses -EV for player and +EV for the casino.

And if the slots really are truly random, the past results doesn't have anything to do with the future EV.

Nominated The Following User Has Nominated This Post:
Casinomeister (Today)
I would think it's a good point if The Casinomeister Nominated it.:clap:

#### Jufo

##### Three-toed sloth
Suppose I deposited 50 get 55 bonus, wager 3150 at 5%.

Should I never play there, because I'm starting at -ev which I can never get back?

One correct point in Enzo's first post was the sentence "For as long as 5% of your totalstake is lower than your total bonus, you are playing at positive EV."

So the bonus you mentioned above can be made +EV playing it so that on average you wager 55 / 5% = 1100 or less. So if you play the bonus with large initial bets in a manner that you have at least 65% chance of busting your bankroll at the beginning (so that you finish the WR only 35% of time), then even a bad bonus like this will actually have positive expectation.

3Dice said:
Looking at it from that point of view one could state that the only way to lose the EV of a bonus is by replacing it with entertainment.

On the other hand if you only play with positive expectation you get much more action with the same initial bankroll and I think you get more entertainment that way. Also many people feel that winning from casinos is more entertaining than losing to them as there is this "I beat the house" psychological reward feeling. Playing bonuses in a manner that they have positive EV also requires some strategical thinking, which might give a new dimension to the otherwise tedious and repetetive casino games, and it actually gives the player some level of control in a situation where he normally has no control over the outcomes dealt to him.

#### blankley

##### Dormant account
'They will be denied, declined, or revoked from players who are deemed to abuse the spirit of this offer. '

So that is a redundant term?

Entertainment can include winning as well as losing?

Entertainment can include a few large bets as well as a lot of small bets?

I fear that casino operators do not accept this as a valid.

#### Jufo

##### Three-toed sloth
'They will be denied, declined, or revoked from players who are deemed to abuse the spirit of this offer. '

So that is a redundant term?

Entertainment can include winning as well as losing?

Entertainment can include a few large bets as well as a lot of small bets?

I fear that casino operators do not accept this as a valid.

That quoted term is against Casinomeister policies (see http://www.casinomeister.com/accredited_casinos.php) so you should avoid playing at all casinos with such unfair terms. I think casinos should accept the fact that offering bonuses have the potential to cause them losses and if they are not willing to take this then they should simply stop offering bonuses instead of revoking such rogue FU clauses.