rumba434
rumba434
  • Threads: 10
  • Posts: 34
Joined: May 13, 2020
August 23rd, 2020 at 5:39:56 AM permalink
If 37 players were logged in to the same single zero roulette table and they all flat bet $1 every turn on their favourite number, each one different, how many turns average would it take before they all had less money than they started with?
ChumpChange
ChumpChange
  • Threads: 128
  • Posts: 5075
Joined: Jun 15, 2018
August 23rd, 2020 at 7:54:31 AM permalink
I used to play an electronic double zero wheel last century. It never really let me win more than 3 bets ahead on even money bets, same with the electronic dollar BJ machines.
ThatDonGuy
ThatDonGuy
  • Threads: 122
  • Posts: 6594
Joined: Jun 22, 2011
Thanked by
ChumpChange
August 23rd, 2020 at 8:43:26 AM permalink
I get 40,500 in simulation.

It becomes 18,750 on a double-zero wheel, and 11,100 on "the devil's wheel" (triple-zero)

OOPS - those were only paying 34-1. See below for corrected numbers.

Bonus math question: how many would it take if there were no zeroes on the wheel?

It would never get to the point where every number was behind, since, in order for that to happen, the sum of the profits would have to be negative, since every individual profit was negative, but on a no-zero wheel, every spin loses 35 on the losing numbers but wins 35 on the winning number, so the sum of the profits is always zero.

Last edited by: ThatDonGuy on Aug 23, 2020
sabre
sabre
  • Threads: 3
  • Posts: 1172
Joined: Aug 16, 2010
Thanked by
RogerKint
August 23rd, 2020 at 2:19:26 PM permalink
Quote: ChumpChange

I used to play an electronic double zero wheel last century. It never really let me win more than 3 bets ahead on even money bets, same with the electronic dollar BJ machines.



Great post. Very on topic. Would read again 10/10
charliepatrick
charliepatrick
  • Threads: 39
  • Posts: 2998
Joined: Jun 17, 2011
August 23rd, 2020 at 5:42:29 PM permalink
I get totally different numbers, so possibly have a bug in my code.
Average length of run before a spin where everyone is broke at the same time = 168346 (average of different runs had 167016-170128, actual runs went from 48420-493344)
Average length of run, where broke people drop out and don't play on, before everyone has goes broke is quite a range 992 (average of diffrent runs 641-1506, actual runs from 37-109513).

In the second analysis (I added the logic before thinking about it much) the person who found the first number stays in, but everyone else went broke. So most often after 37 spins that person goes broke. It's interesting that starting with +35 that the person usually lasts nearly 1000 spins, but sometimes much longer. Thus on normal runs, after say three spins, three people might be up +33 or one doing really quite well. So they're going to last quite a while; with a chance that someone else has a run of luck. Though I'm surprised it's so long.
ThatDonGuy
ThatDonGuy
  • Threads: 122
  • Posts: 6594
Joined: Jun 22, 2011
Thanked by
charliepatrick
August 23rd, 2020 at 6:11:39 PM permalink
Quote: charliepatrick

I get totally different numbers, so possibly have a bug in my code.


I had the bug in my code - I was only paying 34-1.

At 35-1, I get:
170,000 spins on a single-zero wheel
43,500 spins on a double-zero wheel
21,000 spins on a triple-zero wheel
charliepatrick
charliepatrick
  • Threads: 39
  • Posts: 2998
Joined: Jun 17, 2011
August 24th, 2020 at 9:10:22 AM permalink
I did an overnight run and got an average of 169423 (26497-995421).
Ace2
Ace2
  • Threads: 32
  • Posts: 2706
Joined: Oct 2, 2017
August 25th, 2020 at 10:12:42 PM permalink
This problem got me thinking about a somewhat similar problem

If you start rolling a standard die and record the results, how many rolls will it take, on average, for all six numbers to have an equal number of hits?
It’s all about making that GTA
ThatDonGuy
ThatDonGuy
  • Threads: 122
  • Posts: 6594
Joined: Jun 22, 2011
August 26th, 2020 at 6:38:23 AM permalink
Quote: Ace2

This problem got me thinking about a somewhat similar problem

If you start rolling a standard die and record the results, how many rolls will it take, on average, for all six numbers to have an equal number of hits?


I have a feeling the number approaches positive infinity.

Change the problem from a 6-sided die to the toss of a coin.
Let E(n) be the expected number of tosses needed to reach equality where one side has been tossed n times more than the other.
E(0) = 0
E(n) = 1 + 1/2 E(n-1) + 1/2 E(n+1) for n > 0
The starting point is 1 + E(1), since the first toss will result in one side being tossed 1 time more than the other.
Without getting into math details, E(n) = n + n / (n + 1) x E(n+1)
E(1) = 1 + 1/2 E(2)
= 1 + 1/2 (2 + 2/3 E(3)) = 2 + 1/3 E(3)
= 2 + 1/3 (3 + 3/4 E(4)) = 3 + 1/4 E(4)
= 3 + 1/4 (4 + 4/5 E(5)) = 4 + 1/5 E(5)
..., which diverges
Last edited by: ThatDonGuy on Aug 26, 2020
DJTeddyBear
DJTeddyBear
  • Threads: 209
  • Posts: 11049
Joined: Nov 2, 2009
August 26th, 2020 at 10:08:36 AM permalink
I take a different approach to this question.

Let's assume for a moment, that there is one number that is a complete loser. I.E. It never hits. In that case, similar to ThatDonGuy's argument that with a non zero wheel, at least one of the other numbers must be positive, or at least not all numbers be negative.

With that in mind, I choose to calculate the odds of a specific number not hitting in X number of spins. The formula for that is simple:

( 36 / 37 ) ^ X

The chart below shows the results.

Interestingly, it takes 26 spins for a number to pass the 50% mark.

Obviously, it never gets all the way to zero, but gets exceedingly close.

It takes 25,855 spins before it exceeds Excel's capabilities and incorrectly returns a value of zero.


Spins
Odds
1
0.972 972 972 973
2
0.946 676 406 136
3
0.921 090 557 321
4
0.896 196 217 934
5
0.871 974 698 531
10
0.760 339 874 878
25
0.504 103 157 284
26
0.490 478 747 628
50
0.254 119 993 184
75
0.128 102 690 893
100
0.064 576 970 936
150
0.016 410 299 414
200
0.004 170 185 175
300
0.000 269 297 927
400
0.000 017 390 444
500
0.000 001 123 022
750
1.190 097 E-009
1,000
1.261 179 E-012
2,000
1.590 572 E-024
3,000
2.005 996 E-036
4,000
2.529 920 E-048
5,000
3.190 682 E-060
7,500
5.699 349 E-090
10,000
1.018 045 E-119
12,500
1.818 481 E-149
15,000
3.248 258 E-179
17,500
5.802 194 E-209
20,000
1.036 416 E-238
22,500
1.851 296 E-268
25,000
3.306 873 E-298
25,200
1.379 027 E-300
25,400
5.750 799 E-303
25,600
2.398 190 E-305
25,800
1.000 089 E-307
25,825
5.041 483 E-308
25,850
2.541 427 E-308
25,851
2.472 740 E-308
25,852
2.405 909 E-308
25,853
2.340 885 E-308
25,854
2.277 618 E-308
25,855
0
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ ————————————————————————————————————— Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 26th, 2020 at 10:18:02 AM permalink
Quote: Ace2

If you start rolling a standard die and record the results, how many rolls will it take, on average, for all six numbers to have an equal number of hits?



Infinity.

Even with just a coin, the expected number of flips for the number heads and tails to be equal is infinity.

That paradoxical thing is that the probability they will ever be the same is 1. If it will eventually happen, how can it take an infinite number of flips?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
ThatDonGuy
ThatDonGuy
  • Threads: 122
  • Posts: 6594
Joined: Jun 22, 2011
August 26th, 2020 at 12:02:21 PM permalink
Quote: DJTeddyBear

It takes 25,855 spins before it exceeds Excel's capabilities and incorrectly returns a value of zero.


There is a way around it.
Note that log ((36/37)^N) = N (log 36 - log 37), where log is the base 10 logarithm
If the value = a + b, where a is an integer and 0 <= b < 1, then (36/37)^N = 10^a 10^b
You can calculate a and b using the floor() function

26,0004.170 558 E-310
27,0005.259 820 E-322
28,0006.633 574 E-334
29,0008.366 124 E-346
30,0001.055 118 E-357
40,0001.074 158 E-476
50,0001.093 541 E-595
60,0001.113 274 E-714
70,0001.133 363 E-833
80,0001.153 815 E-952
90,0001.174 635 E-1071
100,0001.195 832 E-1190
200,0001.430 013 E-2380
300,0001.710 055 E-3570
400,0002.044 938 E-4760
500,0002.445 401 E-5950
600,0002.924 288 E-7140
700,0003.496 956 E-8330
800,0004.181 771 E-9520
900,0005.000 694 E-10,710
1,000,0005.979 988 E-11,900
2,000,0003.576 025 E-23,799
3,000,0002.138 459 E-35,698
4,000,0001.278 796 E-47,597
5,000,0007.647 183 E-59,497
6,000,0004.573 006 E-71,396
7,000,0002.734 652 E-83,295
8,000,0001.635 318 E-95,194
9,000,0009.779 184 E-107,094
10,000,0005.847 940 E-118,993
Ace2
Ace2
  • Threads: 32
  • Posts: 2706
Joined: Oct 2, 2017
August 26th, 2020 at 4:21:33 PM permalink
Quote: Wizard

Infinity.

Even with just a coin, the expected number of flips for the number heads and tails to be equal is infinity.

That paradoxical thing is that the probability they will ever be the same is 1. If it will eventually happen, how can it take an infinite number of flips?

You refreshed my memory.

I’ve seen the case for a random walk where there’s a 50/50 chance of going left or right. It can be proven that there’s a 100% chance of eventually returning to the origin, but potentially infinite waiting time. Same concept as the coin flip..

Thanks for the replies
It’s all about making that GTA
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 26th, 2020 at 7:03:24 PM permalink
Quote: Ace2

You refreshed my memory.

I’ve seen the case for a random walk where there’s a 50/50 chance of going left or right. It can be proven that there’s a 100% chance of eventually returning to the origin, but potentially infinite waiting time. Same concept as the coin flip..

Thanks for the replies



You're welcome.

Funny how many paradoxes come up when it comes to infinity.

I still say infinity is more of a philosophical concept than a number and claim there is nothing in the physical universe that is infinite in nature.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Ace2
Ace2
  • Threads: 32
  • Posts: 2706
Joined: Oct 2, 2017
August 26th, 2020 at 11:46:48 PM permalink
Quote: Wizard


I still say infinity is more of a philosophical concept than a number and claim there is nothing in the physical universe that is infinite in nature.

So you believe space and time come to an end somewhere/sometime?

Much of calculus is based on infinity
It’s all about making that GTA
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 27th, 2020 at 4:46:57 AM permalink
Quote: Ace2

So you believe space and time come to an end somewhere/sometime?



I believe space does. Time is a tougher one.

Quote:

Much of calculus is based on infinity



I, of course, know that. However, calculus is not a physical thing.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 27th, 2020 at 4:46:57 AM permalink
Quote: Ace2

So you believe space and time come to an end somewhere/sometime?



I believe space does. Time is a tougher one.

Quote:

Much of calculus is based on infinity



I, of course, know that. However, calculus is not a physical thing.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 27th, 2020 at 4:47:09 AM permalink
Quote: Ace2

So you believe space and time come to an end somewhere/sometime?



I believe space does. Time is a tougher one.

Quote:

Much of calculus is based on infinity



I, of course, know that. However, calculus is not a physical thing.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 27th, 2020 at 4:47:10 AM permalink
Quote: Ace2

So you believe space and time come to an end somewhere/sometime?



I believe space does. Time is a tougher one.

Quote:

Much of calculus is based on infinity



I, of course, know that. However, calculus is not a physical thing.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
DJTeddyBear
DJTeddyBear
  • Threads: 209
  • Posts: 11049
Joined: Nov 2, 2009
August 27th, 2020 at 4:57:32 AM permalink
Quote: ThatDonGuy

There is a way around it.
Note that log ((36/37)^N) = N (log 36 - log 37), where log is the base 10 logarithm
If the value = a + b, where a is an integer and 0 <= b < 1, then (36/37)^N = 10^a 10^b
You can calculate a and b using the floor() function

I’m barely an arithmetic guy. Calculus? Fuggedanoudit.


On the other hand...
Quote:

10,000,0005.847 940 E-118,993

I’m a little surprised you didn’t continue this till the exponent got too big for Excel - or became got expressed as scientific notation itself.
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ ————————————————————————————————————— Superstitions are silly, childish, irrational rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
ThatDonGuy
ThatDonGuy
  • Threads: 122
  • Posts: 6594
Joined: Jun 22, 2011
August 27th, 2020 at 6:48:41 AM permalink
Quote: DJTeddyBear

I’m barely an arithmetic guy. Calculus? Fuggedanoudit.


No calculus involved - just logarithms and exponents.
Quote: DJTeddyBear

I’m a little surprised you didn’t continue this till the exponent got too big for Excel - or became got expressed as scientific notation itself.


It gets to the point where 10^N spins has a value of 1 E(-1.18992232997077 E(N-2)).

Pedantic mode: I don't use Excel - I use LibreOffice.
Ace2
Ace2
  • Threads: 32
  • Posts: 2706
Joined: Oct 2, 2017
August 27th, 2020 at 10:57:00 AM permalink
Quote: Wizard

I believe space does.



If there’s a wall where space ends, then the question becomes: what’s behind it. Nothing is probably as abstract as infinity
It’s all about making that GTA
unJon
unJon
  • Threads: 16
  • Posts: 4726
Joined: Jul 1, 2018
August 27th, 2020 at 10:59:23 AM permalink
Quote: Ace2

If there’s a wall where space ends, then the question becomes: what’s behind it. Nothing is probably as abstract as infinity



Or it could be curved in a way that makes it finite.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Freddy911
Freddy911
  • Threads: 0
  • Posts: 4
Joined: Aug 26, 2020
August 27th, 2020 at 11:59:44 AM permalink
Quote: Wizard

You're welcome.

Funny how many paradoxes come up when it comes to infinity.

I still say infinity is more of a philosophical concept than a number and claim there is nothing in the physical universe that is infinite in nature.


I don't expect God or whatever to create a Heaven or whatever that no one will, ever, attain. How about a higher plane of thought? Who was it that said, "Some talk about what is, and, ask why, but, I talk about what isn't, and, ask, why not?" Maybe, a true love of numbers and music, life and soul, well, is what leads to a real and actual infinity. After all, we aren't here for someone else to "fill" us up.
Wizard
Administrator
Wizard
  • Threads: 1508
  • Posts: 26886
Joined: Oct 14, 2009
August 27th, 2020 at 4:42:02 PM permalink
Quote: Ace2

If there’s a wall where space ends, then the question becomes: what’s behind it. Nothing is probably as abstract as infinity



Good point. It might be that you wrap around, like a sphere. This is not a topic I can take very far.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
unJon
unJon
  • Threads: 16
  • Posts: 4726
Joined: Jul 1, 2018
August 27th, 2020 at 9:03:31 PM permalink
Quote: Wizard

Good point. It might be that you wrap around, like a sphere. This is not a topic I can take very far.

You could if you wanted to. There’s a neat explication if quantum mechanics that reduces it to higher order statistical mechanics. I’ll bet you are fluent enough with the latter that you could dive pretty far into the latter.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
  • Jump to: