What are the odds? (Another Roulette Question)

If I am playing single zero roulette, flat betting $100 on 0 and I have a $10000 bankroll, what are the odds of hitting 20k vs going bust?

what if I have a 15k bankroll and I want to hit 30k? 20k to 40k?

Also how do you calculate something like that?

If there is a promotion were you play Single zero roulette until you win or lose 200 units, but if you lose the 200 units you get 15% of your loss rebated back to you, is this positive EV? (Maximizing Variance by flat betting on a single number for 1 unit)

Loss rebate of 15% if player loses 200 units.
House edge = 2.7027%
Std Dev = 5.8378

Here are the house edges for the various numbers of rounds:

500 rounds, H/A = 2.109%
1000 rounds, H/A = 1.920%
10000 rounds, H/A = 2.134%
1000000 rounds, H/A = 2.297% (equals 85% of 2.7027%).

The minimum house edge is attained by playing approximately 1380 rounds. In that case, the H/A is 1.9011%.

Quote: Bondy3

If I am playing single zero roulette, flat betting $100 on 0 and I have a $10000 bankroll, what are the odds of hitting 20k vs going bust?
what if I have a 15k bankroll and I want to hit 30k? 20k to 40k?

100k sim results
$100 bets on 0
35 to 1 payout
Double Bankroll or Bust trying

10k to $20k
($10,000 bankroll with $100 bets gives a 100 unit bankroll)
fail: 56529, 56.53%
success: 43471, 43.47%
(exact calculations to 4 decimals: 43.2285%) From Doctrine of Chances book by Stewart Ethier
avg # of spins: 321

15k to $30k
($15,000 bankroll with $100 bets gives a 150 unit bankroll)
fail: 57956, 57.96%
success: 42044, 42.04%
avg # of spins: 692

20k to $40k
($20,000 bankroll with $100 bets gives a 200 unit bankroll)
fail: 58999, 59.00%
success: 41001, 41.00%
avg # of spins: 1207


If table limits would allow up to say max $300 straight-up bets, Bold Play would be a better way to play and faster, to give one the highest chance of hitting your win goal.
Bold Play being bet just enough to hit your win goal on every spin.
using bold play and $300 max bets on one number you would be successful
20k to $40k
[1] 0.4900659

I would have to re-do my calculations with lower max bets at $200
(should be between 49% and 41%)

Quote: Bondy3

Also how do you calculate something like that?

I would simulate it as I did.
For exact calculations

Stewart Either in his excellent and math heavy Doctrine of Chances book (p 468)
gives the formulas and how to adjust them for each number of numbers bet.
Looks very challenging to say the least.


Good Luck
would love to watch the action

Would going the Bold Play route always give you 23-24 spins before going broke, if your goal was to double up? I just threw some figures into a spreadsheet, and that's the way it looks to me. I'm probably oversimplifying things...

Example:

$500 bankroll, $1000 target

First bet $15. Win gives you 485 + (15*35) = $1010. Loss leaves you at $485.
2nd bet $16. Win gives you 469 + (15*35) = $1029. Loss leaves you at $469.
3rd bet $15. Win gives you 452 + (15*35) = $1013. Loss leaves you at $453.
Etc
Etc
Etc

Quote: 7craps

Stewart Either in his excellent and math heavy Doctrine of Chances book (p 468) gives the formulas and how to adjust them for each number of numbers bet. Looks very challenging to say the least.

Let f(x) be a normal curve with EXP_WIN and STD_DEV for the given bet and given number of rounds played. If the rebate is based on a value y, such that if x < y, then there's a rebate of Z%, then define

g(x) = f(x), if x >= y
g(x) = Z*f(x), if x < y

Then integrate g(x) numerically.

Quote: Scot

Example:

$500 bankroll, $1000 target

First bet $15. Win gives you 485 + (15*35) = $1010. Loss leaves you at $485.
2nd bet $16. Win gives you 469 + (15*35) = $1029. Loss leaves you at $469.
3rd bet $15. Win gives you 452 + (15*35) = $1013. Loss leaves you at $453.
Etc
Etc
Etc
Would going the Bold Play route always give you 23-24 spins before going broke, if your goal was to double up?
I just threw some figures into a spreadsheet, and that's the way it looks to me. I'm probably oversimplifying things...

Real close because to lose 23-24 spins in a row has a higher probability than winning the one time needed to hit the win goal.
If you make a column for probabilities in your spreadsheet you can easily calculate the average # of spins for success or ruin.
I have an Excel sheet for this (I am updating it to handle max bet sizes)

Look again at your numbers. I get
First bet $15. Win gives you 485 + (15*35 + 15 - one gets the winning bet back) = $1025. Loss leaves you at $485. Next bet = 515/35 = 14.71 roundup
2nd bet $15. Win gives you 470 + (15*35 + 15) = $1010. Loss leaves you at $470. Next bet = 530/35 = 15.14 roundup
3rd bet $16. Win gives you 454 + (16*35 + 16) = $1030. Loss leaves you at $454. Next bet = 546/35 = 15.60 roundup
Etc
Etc

thanks for running the sim, I was able to use that to figure out the expected result of the promotion and I found that it is not profitable, even if you only do it with 100 units

what program did you use to run the sim?

Quote: 7craps

100k sim results
$100 bets on 0
35 to 1 payout
Double Bankroll or Bust trying

10k to $20k
($10,000 bankroll with $100 bets gives a 100 unit bankroll)
fail: 56529, 56.53%
success: 43471, 43.47%
(exact calculations to 4 decimals: 43.2285%) From Doctrine of Chances book by Stewart Ethier
avg # of spins: 321

15k to $30k
($15,000 bankroll with $100 bets gives a 150 unit bankroll)
fail: 57956, 57.96%
success: 42044, 42.04%
avg # of spins: 692

20k to $40k
($20,000 bankroll with $100 bets gives a 200 unit bankroll)
fail: 58999, 59.00%
success: 41001, 41.00%
avg # of spins: 1207


If table limits would allow up to say max $300 straight-up bets, Bold Play would be a better way to play and faster, to give one the highest chance of hitting your win goal.
Bold Play being bet just enough to hit your win goal on every spin.
using bold play and $300 max bets on one number you would be successful
20k to $40k
[1] 0.4900659

I would have to re-do my calculations with lower max bets at $200
(should be between 49% and 41%)
I would simulate it as I did.
For exact calculations

Stewart Either in his excellent and math heavy Doctrine of Chances book (p 468)
gives the formulas and how to adjust them for each number of numbers bet.
Looks very challenging to say the least.


Good Luck
would love to watch the action

Ok I spend some more time looking at your sim results and think there is smoething that doesn't feel right

you said that with 66 units you would have a 49% chance of success, but with 100 units it would drop down to 43% and then with 200 units it would drop down to 41% Those numbers don't seem logical, I feel like the jump from 49% to 43% is too sharp for only an additional 33 units,

perhaps something is wrong with your sim?

Quote: Bondy3

Ok I spend some more time looking at your sim results and think there is something that doesn't feel right

you said that with 66 units you would have a 49% chance of success, but with 100 units it would drop down to 43% and then with 200 units it would drop down to 41% Those numbers don't seem logical, I feel like the jump from 49% to 43% is too sharp for only an additional 33 units,

perhaps something is wrong with your sim?

No, nothing wrong with the sims.
The 100 units and above was Flat Betting.
The 66 units you are commenting on is not using flat bets (equal bets).
It is using the concept of Bold Play.

"A gambler with a fixed goal is said to use Bold Play if at each coup he bets
his entire fortune or just enough to achieve his goal in the event of a win,
whichever amount is smaller."

Bold Play actually requires bets from $286 to $554 with a $10k stake (using your example)
The first 10 bets for a $10k stake (all bets lose)
286
294
303
311
320
329
339
349
359
369
and
from $572 to $1109 with a $20k stake.
The first 10 bets for a $20k stake (all bets lose)
572
588
605
622
640
658
677
697
716
737
The above wagers may be more than the house limit would allow in most casinos.
(of course there are ways to get around a one bet max limit by using Full completes/maximums for example)
see
http://en.wikipedia.org/wiki/Roulette#Full_completes.2Fmaximums
That would take some different calculations and simulations.

The example I used at first to double a $20k stake using $300 bets is actually this (I was way off on the $300 bet)
First bet is $572 wins 20,020
with A loss bankroll is 19,428 and we are now 20,572 away from our win target
2nd bet is $588 wins 20,580
with A loss bankroll is 18,840 and we are now 21,160 away from our win target
and so on
A spreadsheet does the calculations nicely for bold play (with correct cell formulas)

A 66 unit bankroll success to double stake by flat betting is about 43.23%
Units: Success prob %
50: 42.91%
25: 41.04%
10: 23.97%
These should be very easy to see by losing 10 in a row,25 in a row etc.
Also to note that the win % (success rate to double) is not linear.

I do simulations in Excel, WinStats and the free program R
Two programs I use to check my Excel work are (they are both free)
1) Gambler's Odds 1.3
(this program limits number of rounds to 3000 using its' Success probability)
2) WinStats by Peanut Software, Gambler's Ruin Simulation
(I helped a bit with the coding and testing of many simulations in WinStats)
RIP Rick Parris

Good Luck

thanks for explaining, I had thought of betting like that but never really done any calculations of it, its nice to see how the numbers work out :)

Also complete bets are a totally different animal, its a semi way to get around the table limit by betting on a spot and all of the neighbors, a few weeks ago there was someone who made a $80,000 complete bet at the Wynn and hit his number. I can only dream of what I would do with that kind of winnings