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.

Quote:ChumpChangeI 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

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.

Quote:charliepatrickI 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

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?

Quote:Ace2This 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

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 |

Quote:Ace2If 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?

Quote:DJTeddyBearIt 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

26,000 | 4.170 558 E-310 |

27,000 | 5.259 820 E-322 |

28,000 | 6.633 574 E-334 |

29,000 | 8.366 124 E-346 |

30,000 | 1.055 118 E-357 |

40,000 | 1.074 158 E-476 |

50,000 | 1.093 541 E-595 |

60,000 | 1.113 274 E-714 |

70,000 | 1.133 363 E-833 |

80,000 | 1.153 815 E-952 |

90,000 | 1.174 635 E-1071 |

100,000 | 1.195 832 E-1190 |

200,000 | 1.430 013 E-2380 |

300,000 | 1.710 055 E-3570 |

400,000 | 2.044 938 E-4760 |

500,000 | 2.445 401 E-5950 |

600,000 | 2.924 288 E-7140 |

700,000 | 3.496 956 E-8330 |

800,000 | 4.181 771 E-9520 |

900,000 | 5.000 694 E-10,710 |

1,000,000 | 5.979 988 E-11,900 |

2,000,000 | 3.576 025 E-23,799 |

3,000,000 | 2.138 459 E-35,698 |

4,000,000 | 1.278 796 E-47,597 |

5,000,000 | 7.647 183 E-59,497 |

6,000,000 | 4.573 006 E-71,396 |

7,000,000 | 2.734 652 E-83,295 |

8,000,000 | 1.635 318 E-95,194 |

9,000,000 | 9.779 184 E-107,094 |

10,000,000 | 5.847 940 E-118,993 |

You refreshed my memory.Quote:WizardInfinity.

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?

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

Quote:Ace2You 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.

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

Much of calculus is based on infinity

Quote:Ace2So 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.

Quote:Ace2So 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.

Quote:Ace2So 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.

Quote:Ace2So 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.

I’m barely an arithmetic guy. Calculus? Fuggedanoudit.Quote:ThatDonGuyThere 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

On the other hand...

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.Quote:

10,000,000 5.847 940 E-118,993

Quote:DJTeddyBearI’m barely an arithmetic guy. Calculus? Fuggedanoudit.

No calculus involved - just logarithms and exponents.

Quote:DJTeddyBearI’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.

Quote:WizardI 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

Quote:Ace2If 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.

Quote:WizardYou'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.

Quote:Ace2If 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.

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