How to calculate expected return for a serie of spins?

Expected return is calculated easly for first spin.
how do I calcluate possible expected return for n-th loost spin.

Let's say what is expected return for 6-th lost spin.

In martingale system this is calculated by the formula:
= 1 - 2 ^ [Spin] * ProbLose ^ [Spin] (where spin is number of a spin and ProbLose is initial probability to loose)

How do we calculate this for some other roulette system where raising bets may be each 3-th or n-th lost spin?


thank you.

codekiddy from europe.

Depends on what you are spinning. Probably, it is -5.26%.

Sorry weaselman, I've posted a blank topic accidentally.

I belive it's not the same cos probability to loose n times is not the same according to this site (with table):

Quote: codekiddy

Sorry weaselman, I've posted a blank topic accidentally.

I belive it's not the same cos probability to loose n times is not the same according to this site (with table):

http://www.gamblecraft.com/review/roulette/martingale.htm

Probability to lose n times is not the same, but the amount you will lose if you do is different too. The differences "cancel out" exactly, so that the expected return does not change.

Thank you for so fast reply,
I meant probability to loose n times in a row is NOT the same (sorry again).

if you look at link I posted you'll se a formula to calculate expected return (in chips) for a serie of lost spins.
Obviously this formula works only for martingale system.
In the table (on that site) look at last two columns and you'll se player edge.
you'll also see that expected return is much less as your losing goes ahead.

Maybe I'm wrong, maybe this is calculation for something else, however I belive it's about expected return because colum G is calculated by dividing column F and E.

Since I can't find an general answer on this site I need someones help to explain this formula and generalize it for all beting systems not only martingale.

thank you so much for your effort.

Quote: codekiddy

Since I can't find an general answer on this site I need someones help to explain this formula and generalize it for all beting systems not only martingale.

Expected return does not depend on the betting system. It is always equal to the house edge. For American roulette, it is -5.26%. This is the fraction of the total amount you bet, that you will be losing on average playing roulette. What (or how often) you bet does not matter.

weaselman,
Your answer is correct and it applies to Expected value, which is not the same as Expected return (for a serie of n lost spins).

That is proven, expected value is allways the same (-5,26% or -2,70%) but expected return is not the same,
it reduces every lost spin!
Did you look at the column E,F,G in the table from the link provided?

Please look and you'll understand and see what am I talking about :)

Kind regards sir!

EDIT:
Ok, I found the answer on this site lol,
it's:
Expected retrun for n spin = Win * (1 - probabilityToLoose ^ spin) - Loss * probabilityToLoose ^ spin

where probability to loose as you see changes from spin to spin and thus exp. return does too.

however I stil don't understand the folowing equation:
Expected value = Expected return / Average bet (according to the link provided)

Here Average bet has nothing to do with real average bet lol,
Can someone explain this? (What everage bet means in this formula and how to calcluate it)
Because I'm geting negative "Average bet" results when using abowe equation in non even-money bets. This is strage because expected value is correct by folowing the above fomula.

Thank you.

Quote: codekiddy

weaselman,
Your answer is correct and it applies to Expected value, which is not the same as Expected return (for a serie of n lost spins).

Yeah, it is the same thing.
The expected value is percentage of your total bet that you are expected to lose. For American wheel is 5.26%.
I am not sure what you mean by "expected return" (it really is a synonym). Perhaps, you are talking about the total amount (as opposed to a percentage) lost after N spins? That's easy to calculate.
After N spins you will have bet 2^(N-1) units. So, your expected loss is 5.26% of that.

Yeah this is Expected return, Expetation of how much money you lose.
but in example of n-th lost spin the probability of n-th lost spin must be taken into account :D

Anyway I understand now.

thank you weaselman for helping me figuring out this,
Your help is much appreciated!

Cheers!

Quote: codekiddy

but in example of n-th lost spin the probability of n-th lost spin must be taken into account :D

It is. It's cooked into that 5.26% figure.

Cokekiddy, if I'm not mistaken (tell me if I am) you're looking at column D on your link and determining that your probability of losing a given spin decreases based on past losing spins. This is not the case. It's showing that the probability of losing N spins, starting at spin #1 decreases as N goes higher. That is true; the chance of losing 10 spins in row is (19/37)^10 = 0.01275%. However, if you've just lost 9 spins in a row the chances of losing the next spin is still 19/37 = 51.35%. That's why column G stays at -2.7%.

Your expected return will always be [Bet] * [House Edge]. There's a few ways of using that. For a single spin of $10 the expected return is $10 * -2.7% = -$0.27. If you bet $5, then $10, then $20, then $40 your average bet for the series is 5+10+20+40 = $75 and your expected return for the series is $75 * -2.7% = -$2.02. To determine the expected value for a system simply find the average bet for a system and multiply it by the house edge. To determine the expected value for a particular series with a system, add all of the bets in the series together and multiply the sum by the house edge.

Thank you for eyplaining this,

But how do you explain column E? ("Average bet") from the link?
By using your formula I'm geting different results then those in a table (column E) from the link.

thats driving me nuts. what mean those numbers from column E? I have to know that otherwise I wont sleep :)

Thank you.

Average bet is the total amount you bet divided over the number of spins. This times column "G" gives column "F" - expected loss per spin. You need to multiply it by number of spins to get the expected total loss (it should be (2^n-1)*0.027 - total bet times house edge).

Ok,
I've done some testings in excel and here is my conclusion (correct me if am worng)

Column E "Average bet" is amount of chips needed multiplied by expected value (column G) so that Expected return will be equal to column F "Bet units" which acctualy IS Expected return :D.

However (lol),
when using this formula for non even-money bets then "Average bet" gets negative because expected return is positive.
so to solve this sign must be changed (for both average bet and expected return) to get right result!
finaly to prove these results, quotient of these result must be Expected value(house edge).

Haha :D
Thank you guys!
If you have some better way/ formula to achive this please advise me!


Kind regards!

OK, so, I just now finally actually went to the webpage in the link. The formulas are in the last line, aren't they? The n-case line tells you exactly how the values were derived...

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Sally

Quote: mustangsally

...This can not be correct.
I have to disagree with this statement and results.
$75 can not be the average bet for the 4 steps.
We need to know the weighted average so we can then multiply it by the house edge...

I understand what you're saying but I think you misunderstood what I was saying due to a very bad slip when I typed that up. Allow me to rephrase it:

If you bet $5 then $10 then $20 then $40 your TOTAL bet for that series is $75. At a house edge of -2.7% the expected return is -$2.02. That is true when looking at that particular series. It's not true, as you stated, when looking at the betting system as a whole or trying to find the average expectation.

I see your point about finding the weighted average bet per spin and appreciate seeing the math.

Quote: mustangsally

We need to know the weighted average so we can then multiply it by the house edge.

This is the sum of the product of each value or the weighted averages (one can use the SUMPRODUCT function in Excel)
or longhand
=$5*1 + $10*0.513513514 + $20*0.263696129 + $40*0.135411525

avg bet = $20.82551873 * (times)
house edge = -1/37
expected value = -0.562851857

Why?
100% of the time we make the first step wager of $5.
The $5 wager loses with a probability of 0.513513514, so we make the $10 wager 0.513513514 of the time
The $10 wager loses with a probability of 0.513513514, so we make the $20 wager 0.513513514*0.513513514 (p^2) of the time
The $20 wager loses with a probability of 0.513513514, so we make the $40 wager 0.513513514*0.513513514*0.513513514 (p^3) of the time
The $40 wager loses with a probability of 0.513513514 and so on

another way to calculate the expected value (return)
=(0.930464352*$5)+(0.069535648*-$75)
Sally

This is freak'n awesome!!
I'm Finaly geting right results in excel :D

Thank you Sally a lot! All your help is much appreciated!

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