Old man betting, interesting way of playing (short term only) + few questions
Your example doesn't quite seem applicable to me to prove your case, because we all agree there is no value of n where a series of n roulette trials has positive EV. The question is about a martingale of unlimited trials. We can't "work backward" from infinity because we can't ever be there.
What would you say is the EV of a martingale with no limits on number of trials, bet size, or bankroll?
Expected value of a single spin: 1*(18/38) - 1(20/38) = -1/19
Expected value of 100 spins: sum 1*(18/38) - 1(20/38) from 1 to 100 = -100/19 which is -1/19 a spin
Expected value of an infinite amount of spins: sum 1*(18/38) - 1(20/38) from 1 to infinity = the sum diverges, the expected value does not exist
Martingale starting at $1:
Expected value of the first spin: 1*(18/38) - 1(20/38) = -1/19
Expected value of the second spin given that you won the first: 1*(18/38) - 1(20/38) = -1/19
Expected value of the second spin given that lost the first: 2*(18/38) - 2(20/38) = -2/19
Expected value of the third spin given that you won the first and second spins: 1*(18/38) - 1(20/38) = -1/19
Expected value of the third spin given that you won the first and lose the second spin: 2*(18/38) - 2(20/38) = -2/19
Expected value of the third spin given that you lost the first and second spins: 4*(18/38) - 4(20/38) = -4/19
and so on. How you write that mathematically I don't know.
Quote: DiscreteMaths2Flat betting $1:
Expected value of a single spin: 1*(18/38) - 1(20/38) = -1/19
Expected value of 100 spins: sum 1*(18/38) - 1(20/38) from 1 to 100 = -100/19 which is -1/19 a spin
Expected value of an infinite amount of spins: sum 1*(18/38) - 1(20/38) from 1 to infinity = the sum diverges, the expected value does not exist
Martingale starting at $1:
Expected value of the first spin: 1*(18/38) - 1(20/38) = -1/19
Expected value of the second spin given that you won the first: 1*(18/38) - 1(20/38) = -1/19
Expected value of the second spin given that lost the first: 2*(18/38) - 2(20/38) = -2/19
Expected value of the third spin given that you won the first and second spins: 1*(18/38) - 1(20/38) = -1/19
Expected value of the third spin given that you won the first and lose the second spin: 2*(18/38) - 2(20/38) = -2/19
Expected value of the third spin given that you lost the first and second spins: 4*(18/38) - 4(20/38) = -4/19
and so on. How you write that mathematically I don't know.
There is no question that each spin of roulette has negative EV…which is equal to the size of the bet divided by 19. That’s the only thing your math shows.
You are disregarding a very important facet of the infinite Martingale system…
In each of your calculations, you are leaving open the possibility of ending on a losing spin—which cannot happen with infinite money and time. Because you would simply place another bet. Let me reiterate…you will never end on a losing bet.
And because you will never end on a losing bet, your probability of losing is zero. And since your probability of losing is zero, your probability of winning is one.
A winning spin in a Martingale system will always net 1-unit profit. So you would have a 100% chance to win 1-unit. Your EV is +1.
This is easily calculable with real math, as I have done so a few posts ago.
Quote: PeeMcGee
A winning spin in a Martingale system will always net 1-unit profit. So you would have a 100% chance to win 1-unit. Your EV is +1.
With an infinite amount of money, the amount of money you have remains the same after every win. There is zero profit.
Quote: PeeMcGeeYou are disregarding a very important facet of the infinite Martingale system…
In each of your calculations, you are leaving open the possibility of ending on a losing spin—which cannot happen with infinite money and time. Because you would simply place another bet. Let me reiterate…you will never end on a losing bet.
And because you will never end on a losing bet, your probability of losing is zero. And since your probability of losing is zero, your probability of winning is one. . . .
This is easily calculable with real math, as I have done so a few posts ago.
PMG.
Where you are going wrong is in treating Infinity as a number which can be equated and manipulated with simple maths equations. Infinity is something that a series can approach but NEVER reach or equal.
An Infinite bankroll cannot exist by definition. It just can't.
Saying that Martingale will have +ev of 1 unit if you have infinite bankroll is like saying that you can stroke a unicorn at the petting zoo if unicorns are real. When you know damned well that unicorns, by definition, are not real.
I saw a unicorn standing on the side of the road 'bout 3:00 in the morning.
Just 'cause you didn't see it doesn't mean it wasn't there.
Don't know why they don't have them at the petting zoo ;-)
Quote: OnceDearWhere you are going wrong is in treating Infinity as a number which can be equated and manipulated with simple maths equations. Infinity is something that a series can approach but NEVER reach or equal.
I am doing no such thing. The math I am using is completely valid. Ironically, it is the other side of the disagreement whom is treating infinity as a number.
By the way, this series is not approaching infinity. It's a convergent series, and it is equal to 1.
Quote:Quote: IbeatyouracesIt's impossible to have infinity + 1
Quote: TomGWith an infinite amount of money, the amount of money you have remains the same after every win. There is zero profit.
The size of the bankroll does not change the fact that the game will pay out.
If I had infinite money, and you gave me $10—that does not change the fact that you gave me $10. Just because my buying power didn’t change doesn’t negate that fact.
And it is probably beyond the scoop of this argument, but there are different sizes of infinity.
Quote: OnceDearAn Infinite bankroll cannot exist by definition. It just can't.
Which I (and MrGoldenSun) have pointed out many times. To quote myself; ‘the question is not is it possible?. The question is what if it were possible?'.
We are starting with the assumption that an infinite bankroll and infinite time does exists. And from that assumption, we are asking is Martingale positive.
Of course, this is purely theoretical—no one is denying that.
By the way, you could kind of simulate an infinite bankroll in the physical world—for example, if you were allowed to verbally declare any money amount and have it be honored. It’s really the infinite time that is the difficult obstacle.
Quote: OnceDearSaying that Martingale will have +ev of 1 unit if you have infinite bankroll is like saying that you can stroke a unicorn at the petting zoo if unicorns are real. When you know damned well that unicorns, by definition, are not real.
If unicorns are real (and assuming they are stroke-able/tamable), then why wouldn’t we be able to stroke one at a petting zoo?
Also, although highly unlikely, but it is mathematically possible to never win another bet for the rest of eternity.
Quote: IbeatyouracesIf I give you $10 out of my infinity, I now don't have infinity. It's impossible to have infinity - 10.
It’s not impossible. If I take away ten numbers from the set of even numbers, the set will still contain infinitely many elements. And it will be a different set.
You are doing what I was accused of—treating infinity as a number. Which it is not.
Quote: IbeatyouracesAlso, although highly unlikely, but it is mathematically possible to never win another bet for the rest of eternity.
Mathematically, the probability of that is exactly ZERO.
Quote: billryanIf I have an infinite amount of money, how would losing ten dollars or a million dollars change that?
What's the answer to this equation?
Infinity - 1,000,000 = ?
Surely it can't be infinity, correct?
Quote: PeeMcGeeThere is no question that each spin of roulette has negative EV…which is equal to the size of the bet divided by 19. That’s the only thing your math shows.
You are disregarding a very important facet of the infinite Martingale system…
In each of your calculations, you are leaving open the possibility of ending on a losing spin—which cannot happen with infinite money and time. Because you would simply place another bet. Let me reiterate…you will never end on a losing bet.
And because you will never end on a losing bet, your probability of losing is zero. And since your probability of losing is zero, your probability of winning is one.
A winning spin in a Martingale system will always net 1-unit profit. So you would have a 100% chance to win 1-unit. Your EV is +1.
This is easily calculable with real math, as I have done so a few posts ago.
Not to sound rude but I think you have some mistakes on how you think EV is calculated. At no point in this thread did you calculate EV for martingale. Yes you posted a series that does indeed converge to 1 but that is not how you calculate EV.
Quote: PeeMcGeeMathematically, the probability of that is exactly ZERO.
Infinitely incorrect. Close to zero, but NOT zero.
Quote: IbeatyouracesWhat's the answer to this equation?
Infinity - 1,000,000 = ?
Surely it can't be infinity, correct?
Then I never had an infinite amount in the first place.
Quote:Quote: billryanIf I have an infinite amount of money, how would losing ten dollars or a million dollars change that?
Quote: IbeatyouracesWhat's the answer to this equation?
Infinity - 1,000,000 = ?
Surely it can't be infinity, correct?Quote: billryanThen I never had an infinite amount in the first place.
Again, infinity is not a number. But if you have a set with infinitely many elements and took away any finite number of elements, the set will still contain infinitely many elements.
Quote: IbeatyouracesInfinitely incorrect. Close to zero, but NOT zero.
It can be difficult to grasp…but it is in fact zero. In fact it is…
. Which is zero.Quote: DiscreteMaths2Quote: PeeMcGeeThere is no question that each spin of roulette has negative EV…which is equal to the size of the bet divided by 19. That’s the only thing your math shows.
You are disregarding a very important facet of the infinite Martingale system…
In each of your calculations, you are leaving open the possibility of ending on a losing spin—which cannot happen with infinite money and time. Because you would simply place another bet. Let me reiterate…you will never end on a losing bet.
And because you will never end on a losing bet, your probability of losing is zero. And since your probability of losing is zero, your probability of winning is one.
A winning spin in a Martingale system will always net 1-unit profit. So you would have a 100% chance to win 1-unit. Your EV is +1.
This is easily calculable with real math, as I have done so a few posts ago.
Not to sound rude but I think you have some mistakes on how you think EV is calculated. At no point in this thread did you calculate EV for martingale. Yes you posted a series that does indeed converge to 1 but that is not how you calculate EV.
I did—the first post of mine on page 4. I calculated the EV just fine. If you don’t agree, please tell me where you believe the flaw lies. I can make it a little more formal (which it seems you are seeking).
Assuming infinite bankroll and time, a Martingale system will have expected value as follows.
Let W be a discrete random variable (winnings) taking values w0, w1, … with probabilities p0, p1, … respectively. Then the expected value of this random variable is:
Quote: PeeMcGeeThe size of the bankroll does not change the fact that the game will pay out.
If I had infinite money, and you gave me $10—that does not change the fact that you gave me $10. Just because my buying power didn’t change doesn’t negate that fact.
Your original claim was that it had a positive expected value. That is different than saying the game will pay out after a win: https://en.wikipedia.org/wiki/Expected_value
Quote: TomGYour original claim was that it had a positive expected value. That is different than saying the game will pay out after a win: https://en.wikipedia.org/wiki/Expected_value
The game will pay out an expected amount of 1-unit. The probability-weighted average of all possible values is 1. That is the expected value. That has been my claim the entire time. The bettor’s purchasing-power being unchanged does not negate that fact.
I am not sure why you provided that link. Nothing in that article contradicts this claim.
Quote: PeeMcGee...It can be difficult to grasp…but it is in fact zero. In fact it is…
I'll say this again. You are incorrect. If something has a probability of even 0.0000(insert 5,000,000 quadrillion quadrillion quadrillion more zeros)000001% chance of happening, it's still NOT zero.
Quote: billryanIf I have an infinite amount of money, how would losing ten dollars or a million dollars change that?
How would it change your life? Win or lose?
Quote: PeeMcGee
Let W be a discrete random variable (winnings) taking values w0, w1, … with probabilities p0, p1, … respectively. Then the expected value of this random variable is:
Because you are leaving out where you lose money. First spin is 1*(18/38) - 1(20/38) not just 1*(18/38).
Quote: OnceDearHow would it change your life? Win or lose?
I'd tip more on wins.
>>>>a way of increasing my odds for the short term :)
To increase your odds for the short term: CHEAT,
To increase your odds for the longer term: CHEAT LONGER.
There is NO other way to accomplish this except the old and honest way: Buy the Casino.
| # Wagers | EV |
|---|---|
| 1 | -0.052631579 |
| 2 | -0.108033241 |
| 3 | -0.16635078 |
| 4 | -0.227737663 |
| 5 | -0.292355435 |
| 6 | -0.360374142 |
| 7 | -0.431972781 |
| 8 | -0.50733977 |
| 9 | -0.586673442 |
| 10 | -0.67018257 |
| 11 | -0.758086916 |
| 12 | -0.850617806 |
| 13 | -0.948018743 |
| 14 | -1.050546046 |
| 15 | -1.158469522 |
| 16 | -1.272073181 |
| 17 | -1.39165598 |
| 18 | -1.51753261 |
| 19 | -1.650034327 |
| 20 | -1.789509818 |
| 21 | -1.936326124 |
| 22 | -2.090869604 |
| 23 | -2.253546951 |
| 24 | -2.424786265 |
| 25 | -2.605038173 |
| 26 | -2.794777025 |
| 27 | -2.994502131 |
| 28 | -3.204739085 |
| 29 | -3.426041143 |
| 30 | -3.658990676 |
| 31 | -3.904200712 |
| 32 | -4.162316539 |
| 33 | -4.434017409 |
| 34 | -4.720018326 |
| 35 | -5.021071922 |
| 36 | -5.337970444 |
| 37 | -5.671547836 |
| 38 | -6.022681932 |
| 39 | -6.392296771 |
| 40 | -6.781365022 |
| 41 | -7.190910549 |
| 42 | -7.622011105 |
| 43 | -8.075801163 |
| 44 | -8.553474908 |
| 45 | -9.056289377 |
| 46 | -9.585567765 |
| 47 | -10.14270291 |
| 48 | -10.72916096 |
| 49 | -11.34648522 |
| 50 | -11.99630023 |
| 51 | -12.68031603 |
| 52 | -13.40033267 |
| 53 | -14.15824491 |
| 54 | -14.95604728 |
| 55 | -15.79583924 |
| 56 | -16.67983078 |
| 57 | -17.61034819 |
| 58 | -18.5898402 |
| 59 | -19.62088442 |
| 60 | -20.70619412 |
| 61 | -21.84862539 |
| 62 | -23.05118462 |
| 63 | -24.31703645 |
| 64 | -25.64951205 |
| 65 | -27.05211795 |
| 66 | -28.52854521 |
| 67 | -30.08267916 |
| 68 | -31.71860965 |
| 69 | -33.44064173 |
| 70 | -35.25330709 |
| 71 | -37.16137588 |
| 72 | -39.16986935 |
| 73 | -41.284073 |
| 74 | -43.50955052 |
| 75 | -45.85215845 |
| 76 | -48.31806152 |
| 77 | -50.91374897 |
| 78 | -53.64605155 |
| 79 | -56.52215953 |
| 80 | -59.54964161 |
| 81 | -62.73646485 |
| 82 | -66.09101563 |
| 83 | -69.62212172 |
| 84 | -73.33907549 |
| 85 | -77.25165841 |
| 86 | -81.37016675 |
| 87 | -85.70543868 |
| 88 | -90.26888282 |
| 89 | -95.07250824 |
| 90 | -100.128956 |
| 91 | -105.4515327 |
| 92 | -111.0542449 |
| 93 | -116.9518368 |
| 94 | -123.1598282 |
| 95 | -129.694556 |
| 96 | -136.5732168 |
| 97 | -143.8139124 |
| 98 | -151.4356973 |
| 99 | -159.4586287 |
| 100 | -167.9038197 |
| 101 | -176.7934944 |
| 102 | -186.1510468 |
| 103 | -196.0011019 |
| 104 | -206.3695809 |
| 105 | -217.2837694 |
| 106 | -228.7723888 |
| 107 | -240.8656724 |
| 108 | -253.5954447 |
| 109 | -266.9952049 |
| 110 | -281.1002157 |
| 111 | -295.9475955 |
| 112 | -311.5764163 |
| 113 | -328.0278066 |
| 114 | -345.3450596 |
| 115 | -363.5737469 |
| 116 | -382.7618389 |
| 117 | -402.9598304 |
| 118 | -424.2208741 |
| 119 | -446.6009201 |
| 120 | -470.1588633 |
| 121 | -494.9566982 |
| 122 | -521.0596823 |
| 123 | -548.5365077 |
| 124 | -577.4594818 |
| 125 | -607.9047177 |
| 126 | -639.9523344 |
| 127 | -673.6866678 |
| 128 | -709.1964924 |
| 129 | -746.5752552 |
| 130 | -785.9213212 |
| 131 | -827.3382329 |
| 132 | -870.934982 |
| 133 | -916.8262968 |
| 134 | -965.132944 |
| 135 | -1015.982046 |
| 136 | -1069.507417 |
| 137 | -1125.849913 |
| 138 | -1185.157803 |
| 139 | -1247.587161 |
| 140 | -1313.302275 |
| 141 | -1382.476079 |
| 142 | -1455.290609 |
| 143 | -1531.937483 |
| 144 | -1612.618403 |
| 145 | -1697.545688 |
| 146 | -1786.942829 |
| 147 | -1881.045084 |
| 148 | -1980.100088 |
| 149 | -2084.368514 |
| 150 | -2194.124751 |
| 151 | -2309.657633 |
| 152 | -2431.271192 |
| 153 | -2559.285466 |
| 154 | -2694.037332 |
| 155 | -2835.881402 |
| 156 | -2985.19095 |
| 157 | -3142.358895 |
| 158 | -3307.798836 |
| 159 | -3481.946144 |
| 160 | -3665.259099 |
| 161 | -3858.220104 |
| 162 | -4061.336951 |
| 163 | -4275.144159 |
| 164 | -4500.204378 |
| 165 | -4737.109872 |
| 166 | -4986.484076 |
| 167 | -5248.983237 |
| 168 | -5525.298145 |
| 169 | -5816.155942 |
| 170 | -6122.322044 |
| 171 | -6444.602152 |
| 172 | -6783.84437 |
| 173 | -7140.941442 |
| 174 | -7516.833097 |
| 175 | -7912.508523 |
| 176 | -8329.008972 |
| 177 | -8767.430497 |
| 178 | -9228.926839 |
| 179 | -9714.712462 |
| 180 | -10226.06575 |
| 181 | -10764.33237 |
| 182 | -11330.92881 |
| 183 | -11927.34611 |
| 184 | -12555.1538 |
| 185 | -13216.004 |
| 186 | -13911.63579 |
| 187 | -14643.87978 |
| 188 | -15414.66293 |
| 189 | -16226.01361 |
| 190 | -17080.06696 |
| 191 | -17979.07048 |
| 192 | -18925.38998 |
| 193 | -19921.51577 |
| 194 | -20970.06923 |
| 195 | -22073.80972 |
| 196 | -23235.64181 |
| 197 | -24458.62295 |
| 198 | -25745.97153 |
| 199 | -27101.0753 |
| 200 | -28527.50031 |
| 201 | -30029.00033 |
| 202 | -31609.52666 |
| 203 | -33273.23859 |
| 204 | -35024.51431 |
| 205 | -36867.96243 |
| 206 | -38808.43413 |
| 207 | -40851.03593 |
| 208 | -43001.14308 |
| 209 | -45264.41377 |
| 210 | -47646.80397 |
| 211 | -50154.58313 |
| 212 | -52794.35066 |
| 213 | -55573.05333 |
| 214 | -58498.0035 |
| 215 | -61576.89842 |
| 216 | -64817.84045 |
| 217 | -68229.35836 |
| 218 | -71820.42986 |
| 219 | -75600.50511 |
| 220 | -79579.5317 |
| 221 | -83767.98073 |
| 222 | -88176.87446 |
| 223 | -92817.81522 |
| 224 | -97703.01602 |
| 225 | -102845.3327 |
| 226 | -108258.2975 |
| 227 | -113956.1553 |
| 228 | -119953.9003 |
| 229 | -126267.3161 |
| 230 | -132913.017 |
| 231 | -139908.4915 |
| 232 | -147272.149 |
| 233 | -155023.3674 |
| 234 | -163182.5446 |
| 235 | -171771.1522 |
| 236 | -180811.7918 |
| 237 | -190328.2545 |
| 238 | -200345.5837 |
| 239 | -210890.1407 |
| 240 | -221989.6745 |
| 241 | -233673.3942 |
| 242 | -245972.0465 |
| 243 | -258917.9963 |
| 244 | -272545.3119 |
| 245 | -286889.8546 |
| 246 | -301989.3733 |
| 247 | -317883.6035 |
| 248 | -334614.3721 |
| 249 | -352225.7074 |
| 250 | -370763.9552 |
| 251 | -390277.9002 |
| 252 | -410818.895 |
| 253 | -432440.9947 |
| 254 | -455201.0997 |
| 255 | -479159.1049 |
| 256 | -504378.0578 |
| 257 | -530924.324 |
| 258 | -558867.7621 |
| 259 | -588281.9075 |
| 260 | -619244.1658 |
| 261 | -651836.0166 |
| 262 | -686143.228 |
| 263 | -722256.0821 |
| 264 | -760269.6128 |
| 265 | -800283.8555 |
| 266 | -842404.1111 |
| 267 | -886741.2222 |
| 268 | -933411.8655 |
| 269 | -982538.8584 |
| 270 | -1034251.483 |
| 271 | -1088685.824 |
| 272 | -1145985.13 |
| 273 | -1206300.19 |
| 274 | -1269789.726 |
| 275 | -1336620.817 |
| 276 | -1406969.334 |
| 277 | -1481020.404 |
| 278 | -1558968.899 |
| 279 | -1641019.946 |
| 280 | -1727389.469 |
| 281 | -1818304.757 |
| 282 | -1914005.06 |
| 283 | -2014742.221 |
| 284 | -2120781.338 |
| 285 | -2232401.461 |
| 286 | -2349896.328 |
| 287 | -2473575.134 |
| 288 | -2603763.352 |
| 289 | -2740803.581 |
| 290 | -2885056.454 |
| 291 | -3036901.583 |
| 292 | -3196738.561 |
| 293 | -3364988.012 |
| 294 | -3542092.696 |
| 295 | -3728518.68 |
| 296 | -3924756.558 |
| 297 | -4131322.746 |
| 298 | -4348760.837 |
| 299 | -4577643.039 |
| 300 | -4818571.673 |
| 301 | -5072180.761 |
| 302 | -5339137.696 |
| 303 | -5620144.996 |
| 304 | -5915942.153 |
| 305 | -6227307.583 |
| 306 | -6555060.666 |
| 307 | -6900063.911 |
| 308 | -7263225.223 |
| 309 | -7645500.287 |
| 310 | -8047895.091 |
| 311 | -8471468.57 |
| 312 | -8917335.389 |
| 313 | -9386668.884 |
| 314 | -9880704.141 |
| 315 | -10400741.25 |
| 316 | -10948148.74 |
| 317 | -11524367.15 |
| 318 | -12130912.84 |
| 319 | -12769381.99 |
| 320 | -13441454.78 |
| 321 | -14148899.82 |
| 322 | -14893578.81 |
| 323 | -15677451.43 |
| 324 | -16502580.51 |
| 325 | -17371137.43 |
| 326 | -18285407.87 |
| 327 | -19247797.81 |
| 328 | -20260839.85 |
| 329 | -21327199.9 |
| 330 | -22449684.16 |
| 331 | -23631246.53 |
| 332 | -24874996.4 |
| 333 | -26184206.79 |
| 334 | -27562322.99 |
| 335 | -29012971.63 |
| 336 | -30539970.18 |
| 337 | -32147337.09 |
| 338 | -33839302.25 |
| 339 | -35620318.21 |
| 340 | -37495071.85 |
| 341 | -39468496.74 |
| 342 | -41545786.1 |
| 343 | -43732406.47 |
| 344 | -46034112.13 |
| 345 | -48456960.19 |
| 346 | -51007326.56 |
| 347 | -53691922.75 |
| 348 | -56517813.48 |
| 349 | -59492435.29 |
| 350 | -62623616.15 |
| 351 | -65919596 |
| 352 | -69389048.47 |
| 353 | -73041103.71 |
| 354 | -76885372.37 |
| 355 | -80931970.97 |
| 356 | -85191548.45 |
| 357 | -89675314.21 |
| 358 | -94395067.64 |
| 359 | -99363229.14 |
| 360 | -104592872.8 |
| 361 | -110097760.9 |
| 362 | -115892380 |
| 363 | -121991979 |
| 364 | -128412609.5 |
| 365 | -135171168 |
| 366 | -142285440 |
| 367 | -149774147.4 |
| 368 | -157656997.3 |
| 369 | -165954734.1 |
| 370 | -174689193.8 |
| 371 | -183883362 |
| 372 | -193561433.7 |
| 373 | -203748877.7 |
| 374 | -214472502.9 |
| 375 | -225760529.4 |
| 376 | -237642662.5 |
| 377 | -250150171.2 |
| 378 | -263315969.7 |
| 379 | -277174705 |
| 380 | -291762847.4 |
| 381 | -307118786.8 |
| 382 | -323282933.5 |
| 383 | -340297824.8 |
| 384 | -358208236.7 |
| 385 | -377061301.8 |
| 386 | -396906633.6 |
| 387 | -417796456.4 |
| 388 | -439785743.7 |
| 389 | -462932361.8 |
| 390 | -487297223 |
| 391 | -512944445.3 |
| 392 | -539941521.5 |
| 393 | -568359496.3 |
| 394 | -598273154.1 |
| 395 | -629761214.9 |
| 396 | -662906542 |
| 397 | -697796360.1 |
| 398 | -734522484.3 |
| 399 | -773181562.5 |
| 400 | -813875329 |
| 401 | -856710872.7 |
| 402 | -901800918.7 |
| 403 | -949264125 |
| 404 | -999225394.8 |
| 405 | -1051816205 |
| 406 | -1107174953 |
| 407 | -1165447319 |
| 408 | -1226786651 |
| 409 | -1291354370 |
| 410 | -1359320389 |
| 411 | -1430863568 |
| 412 | -1506172177 |
| 413 | -1585444397 |
| 414 | -1668888839 |
| 415 | -1756725093 |
| 416 | -1849184309 |
| 417 | -1946509799 |
| 418 | -2048957683 |
| 419 | -2156797561 |
| 420 | -2270313222 |
| 421 | -2389803392 |
| 422 | -2515582518 |
| 423 | -2647981598 |
| 424 | -2787349050 |
| 425 | -2934051632 |
| 426 | -3088475402 |
| 427 | -3251026739 |
| 428 | -3422133410 |
| 429 | -3602245695 |
| 430 | -3791837573 |
| 431 | -3991407972 |
| 432 | -4201482076 |
| 433 | -4422612711 |
| 434 | -4655381802 |
| 435 | -4900401896 |
| 436 | -5158317786 |
| 437 | -5429808196 |
| 438 | -5715587574 |
| 439 | -6016407973 |
| 440 | -6333061024 |
| 441 | -6666380026 |
| 442 | -7017242132 |
| 443 | -7386570666 |
| 444 | -7775337543 |
| 445 | -8184565835 |
| 446 | -8615332458 |
| 447 | -9068771008 |
| 448 | -9546074745 |
| 449 | -10048499732 |
| 450 | -10577368139 |
| 451 | -11134071725 |
| 452 | -11720075500 |
| 453 | -12336921579 |
| 454 | -12986233242 |
| 455 | -13669719202 |
| 456 | -14389178107 |
| 457 | -15146503271 |
| 458 | -15943687653 |
| 459 | -16782829109 |
| 460 | -17666135904 |
| 461 | -18595932531 |
| 462 | -19574665822 |
| 463 | -20604911391 |
| 464 | -21689380412 |
| 465 | -22830926750 |
| 466 | -24032554473 |
| 467 | -25297425762 |
| 468 | -26628869223 |
| 469 | -28030388656 |
| 470 | -29505672269 |
| 471 | -31058602389 |
| 472 | -32693265672 |
| 473 | -34413963865 |
| 474 | -36225225122 |
| 475 | -38131815918 |
| 476 | -40138753597 |
| 477 | -42251319576 |
| 478 | -44475073238 |
| 479 | -46815866567 |
| 480 | -49279859544 |
| 481 | -51873536362 |
| 482 | -54603722486 |
| 483 | -57477602617 |
| 484 | -60502739597 |
| 485 | -63687094313 |
| 486 | -67039046645 |
| 487 | -70567417521 |
| 488 | -74281492128 |
| 489 | -78191044345 |
| 490 | -82306362469 |
| 491 | -86638276283 |
| 492 | -91198185561 |
| 493 | -95998090064 |
| 494 | -1.01051E+11 |
| 495 | -1.06369E+11 |
| 496 | -1.11967E+11 |
| 497 | -1.1786E+11 |
| 498 | -1.24064E+11 |
| 499 | -1.30593E+11 |
| 500 | -1.37467E+11 |
| 501 | -1.44702E+11 |
| 502 | -1.52318E+11 |
| 503 | -1.60334E+11 |
| 504 | -1.68773E+11 |
| 505 | -1.77656E+11 |
| 506 | -1.87006E+11 |
| 507 | -1.96849E+11 |
| 508 | -2.07209E+11 |
| 509 | -2.18115E+11 |
| 510 | -2.29594E+11 |
| 511 | -2.41678E+11 |
| 512 | -2.54398E+11 |
| 513 | -2.67788E+11 |
| 514 | -2.81882E+11 |
| 515 | -2.96718E+11 |
| 516 | -3.12334E+11 |
| 517 | -3.28773E+11 |
| 518 | -3.46077E+11 |
| 519 | -3.64291E+11 |
| 520 | -3.83465E+11 |
| 521 | -4.03647E+11 |
| 522 | -4.24891E+11 |
| 523 | -4.47254E+11 |
| 524 | -4.70794E+11 |
| 525 | -4.95573E+11 |
| 526 | -5.21655E+11 |
| 527 | -5.49111E+11 |
| 528 | -5.78011E+11 |
| 529 | -6.08433E+11 |
| 530 | -6.40456E+11 |
| 531 | -6.74164E+11 |
| 532 | -7.09646E+11 |
| 533 | -7.46996E+11 |
| 534 | -7.86312E+11 |
| 535 | -8.27697E+11 |
| 536 | -8.7126E+11 |
| 537 | -9.17115E+11 |
| 538 | -9.65385E+11 |
| 539 | -1.01619E+12 |
| 540 | -1.06968E+12 |
| 541 | -1.12598E+12 |
| 542 | -1.18524E+12 |
| 543 | -1.24762E+12 |
| 544 | -1.31328E+12 |
| 545 | -1.3824E+12 |
| 546 | -1.45516E+12 |
| 547 | -1.53175E+12 |
| 548 | -1.61237E+12 |
| 549 | -1.69723E+12 |
| 550 | -1.78656E+12 |
| 551 | -1.88059E+12 |
| 552 | -1.97957E+12 |
| 553 | -2.08375E+12 |
| 554 | -2.19342E+12 |
| 555 | -2.30887E+12 |
| 556 | -2.43039E+12 |
| 557 | -2.5583E+12 |
| 558 | -2.69295E+12 |
| 559 | -2.83468E+12 |
| 560 | -2.98388E+12 |
| 561 | -3.14092E+12 |
| 562 | -3.30624E+12 |
| 563 | -3.48025E+12 |
| 564 | -3.66342E+12 |
| 565 | -3.85623E+12 |
| 566 | -4.05919E+12 |
| 567 | -4.27283E+12 |
| 568 | -4.49772E+12 |
| 569 | -4.73444E+12 |
| 570 | -4.98362E+12 |
| 571 | -5.24592E+12 |
| 572 | -5.52202E+12 |
| 573 | -5.81265E+12 |
| 574 | -6.11858E+12 |
| 575 | -6.44061E+12 |
| 576 | -6.77959E+12 |
| 577 | -7.13641E+12 |
| 578 | -7.51201E+12 |
| 579 | -7.90738E+12 |
| 580 | -8.32356E+12 |
| 581 | -8.76164E+12 |
| 582 | -9.22278E+12 |
| 583 | -9.70819E+12 |
| 584 | -1.02191E+13 |
| 585 | -1.0757E+13 |
| 586 | -1.13232E+13 |
| 587 | -1.19191E+13 |
| 588 | -1.25464E+13 |
| 589 | -1.32068E+13 |
| 590 | -1.39019E+13 |
| 591 | -1.46335E+13 |
| 592 | -1.54037E+13 |
| 593 | -1.62144E+13 |
| 594 | -1.70678E+13 |
| 595 | -1.79661E+13 |
| 596 | -1.89117E+13 |
| 597 | -1.99071E+13 |
| 598 | -2.09548E+13 |
| 599 | -2.20577E+13 |
| 600 | -2.32186E+13 |
| 601 | -2.44407E+13 |
| 602 | -2.5727E+13 |
| 603 | -2.70811E+13 |
| 604 | -2.85064E+13 |
| 605 | -3.00067E+13 |
| 606 | -3.1586E+13 |
| 607 | -3.32485E+13 |
| 608 | -3.49984E+13 |
| 609 | -3.68404E+13 |
| 610 | -3.87794E+13 |
| 611 | -4.08204E+13 |
| 612 | -4.29688E+13 |
| 613 | -4.52304E+13 |
| 614 | -4.76109E+13 |
| 615 | -5.01167E+13 |
| 616 | -5.27545E+13 |
| 617 | -5.5531E+13 |
| 618 | -5.84537E+13 |
| 619 | -6.15302E+13 |
| 620 | -6.47686E+13 |
| 621 | -6.81775E+13 |
| 622 | -7.17658E+13 |
| 623 | -7.55429E+13 |
| 624 | -7.95189E+13 |
| 625 | -8.37041E+13 |
| 626 | -8.81096E+13 |
| 627 | -9.27469E+13 |
| 628 | -9.76283E+13 |
| 629 | -1.02767E+14 |
| 630 | -1.08175E+14 |
| 631 | -1.13869E+14 |
| 632 | -1.19862E+14 |
| 633 | -1.26171E+14 |
| 634 | -1.32811E+14 |
| 635 | -1.39801E+14 |
| 636 | -1.47159E+14 |
| 637 | -1.54904E+14 |
| 638 | -1.63057E+14 |
| 639 | -1.71639E+14 |
| 640 | -1.80673E+14 |
| 641 | -1.90182E+14 |
| 642 | -2.00191E+14 |
| 643 | -2.10728E+14 |
| 644 | -2.21819E+14 |
| 645 | -2.33493E+14 |
| 646 | -2.45783E+14 |
| 647 | -2.58718E+14 |
| 648 | -2.72335E+14 |
| 649 | -2.86669E+14 |
| 650 | -3.01756E+14 |
| 651 | -3.17638E+14 |
| 652 | -3.34356E+14 |
| 653 | -3.51954E+14 |
| 654 | -3.70478E+14 |
| 655 | -3.89977E+14 |
| 656 | -4.10502E+14 |
| 657 | -4.32107E+14 |
| 658 | -4.54849E+14 |
| 659 | -4.78789E+14 |
| 660 | -5.03988E+14 |
| 661 | -5.30514E+14 |
| 662 | -5.58436E+14 |
| 663 | -5.87827E+14 |
| 664 | -6.18765E+14 |
| 665 | -6.51332E+14 |
| 666 | -6.85613E+14 |
| 667 | -7.21698E+14 |
| 668 | -7.59682E+14 |
| 669 | -7.99665E+14 |
| 670 | -8.41753E+14 |
| 671 | -8.86055E+14 |
| 672 | -9.3269E+14 |
| 673 | -9.81779E+14 |
| 674 | -1.03345E+15 |
| 675 | -1.08784E+15 |
| 676 | -1.1451E+15 |
| 677 | -1.20537E+15 |
| 678 | -1.26881E+15 |
| 679 | -1.33559E+15 |
| 680 | -1.40588E+15 |
| 681 | -1.47987E+15 |
| 682 | -1.55776E+15 |
| 683 | -1.63975E+15 |
| 684 | -1.72605E+15 |
| 685 | -1.8169E+15 |
| 686 | -1.91252E+15 |
| 687 | -2.01318E+15 |
| 688 | -2.11914E+15 |
| 689 | -2.23067E+15 |
| 690 | -2.34808E+15 |
| 691 | -2.47166E+15 |
| 692 | -2.60175E+15 |
| 693 | -2.73868E+15 |
| 694 | -2.88282E+15 |
| 695 | -3.03455E+15 |
| 696 | -3.19426E+15 |
| 697 | -3.36238E+15 |
| 698 | -3.53935E+15 |
| 699 | -3.72563E+15 |
| 700 | -3.92172E+15 |
| 701 | -4.12812E+15 |
| 702 | -4.34539E+15 |
| 703 | -4.5741E+15 |
| 704 | -4.81484E+15 |
| 705 | -5.06825E+15 |
| 706 | -5.335E+15 |
| 707 | -5.61579E+15 |
| 708 | -5.91136E+15 |
| 709 | -6.22248E+15 |
| 710 | -6.54998E+15 |
| 711 | -6.89472E+15 |
| 712 | -7.2576E+15 |
| 713 | -7.63958E+15 |
| 714 | -8.04166E+15 |
| 715 | -8.46491E+15 |
| 716 | -8.91043E+15 |
| 717 | -9.3794E+15 |
| 718 | -9.87305E+15 |
| 719 | -1.03927E+16 |
| 720 | -1.09397E+16 |
| 721 | -1.15154E+16 |
| 722 | -1.21215E+16 |
| 723 | -1.27595E+16 |
| 724 | -1.3431E+16 |
| 725 | -1.41379E+16 |
| 726 | -1.4882E+16 |
| 727 | -1.56653E+16 |
| 728 | -1.64898E+16 |
| 729 | -1.73577E+16 |
| 730 | -1.82712E+16 |
| 731 | -1.92329E+16 |
| 732 | -2.02451E+16 |
| 733 | -2.13107E+16 |
| 734 | -2.24323E+16 |
| 735 | -2.36129E+16 |
| 736 | -2.48557E+16 |
| 737 | -2.61639E+16 |
| 738 | -2.7541E+16 |
| 739 | -2.89905E+16 |
| 740 | -3.05163E+16 |
| 741 | -3.21224E+16 |
| 742 | -3.38131E+16 |
| 743 | -3.55927E+16 |
| 744 | -3.7466E+16 |
| 745 | -3.94379E+16 |
| 746 | -4.15136E+16 |
| 747 | -4.36985E+16 |
| 748 | -4.59985E+16 |
| 749 | -4.84194E+16 |
| 750 | -5.09678E+16 |
| 751 | -5.36503E+16 |
| 752 | -5.6474E+16 |
| 753 | -5.94464E+16 |
| 754 | -6.25751E+16 |
| 755 | -6.58685E+16 |
| 756 | -6.93353E+16 |
| 757 | -7.29845E+16 |
| 758 | -7.68258E+16 |
| 759 | -8.08693E+16 |
| 760 | -8.51256E+16 |
| 761 | -8.96059E+16 |
| 762 | -9.43219E+16 |
| 763 | -9.92863E+16 |
| 764 | -1.04512E+17 |
| 765 | -1.10012E+17 |
| 766 | -1.15803E+17 |
| 767 | -1.21897E+17 |
| 768 | -1.28313E+17 |
| 769 | -1.35066E+17 |
| 770 | -1.42175E+17 |
| 771 | -1.49658E+17 |
| 772 | -1.57535E+17 |
| 773 | -1.65826E+17 |
| 774 | -1.74554E+17 |
| 775 | -1.83741E+17 |
| 776 | -1.93412E+17 |
| 777 | -2.03591E+17 |
| 778 | -2.14306E+17 |
| 779 | -2.25586E+17 |
| 780 | -2.37459E+17 |
| 781 | -2.49956E+17 |
| 782 | -2.63112E+17 |
| 783 | -2.7696E+17 |
| 784 | -2.91537E+17 |
| 785 | -3.06881E+17 |
| 786 | -3.23033E+17 |
| 787 | -3.40034E+17 |
| 788 | -3.57931E+17 |
| 789 | -3.76769E+17 |
| 790 | -3.96599E+17 |
| 791 | -4.17473E+17 |
| 792 | -4.39445E+17 |
| 793 | -4.62574E+17 |
| 794 | -4.8692E+17 |
| 795 | -5.12547E+17 |
| 796 | -5.39523E+17 |
| 797 | -5.67919E+17 |
| 798 | -5.9781E+17 |
| 799 | -6.29273E+17 |
| 800 | -6.62393E+17 |
| 801 | -6.97256E+17 |
| 802 | -7.33954E+17 |
| 803 | -7.72583E+17 |
| 804 | -8.13245E+17 |
| 805 | -8.56047E+17 |
| 806 | -9.01102E+17 |
| 807 | -9.48529E+17 |
| 808 | -9.98451E+17 |
| 809 | -1.051E+18 |
| 810 | -1.10632E+18 |
| 811 | -1.16454E+18 |
| 812 | -1.22584E+18 |
| 813 | -1.29035E+18 |
| 814 | -1.35827E+18 |
| 815 | -1.42976E+18 |
| 816 | -1.50501E+18 |
| 817 | -1.58422E+18 |
| 818 | -1.6676E+18 |
| 819 | -1.75536E+18 |
| 820 | -1.84775E+18 |
| 821 | -1.945E+18 |
| 822 | -2.04737E+18 |
| 823 | -2.15513E+18 |
| 824 | -2.26855E+18 |
| 825 | -2.38795E+18 |
| 826 | -2.51363E+18 |
| 827 | -2.64593E+18 |
| 828 | -2.78519E+18 |
| 829 | -2.93178E+18 |
| 830 | -3.08608E+18 |
| 831 | -3.24851E+18 |
| 832 | -3.41948E+18 |
| 833 | -3.59946E+18 |
| 834 | -3.7889E+18 |
| 835 | -3.98832E+18 |
| 836 | -4.19823E+18 |
| 837 | -4.41919E+18 |
| 838 | -4.65178E+18 |
| 839 | -4.89661E+18 |
| 840 | -5.15432E+18 |
| 841 | -5.4256E+18 |
| 842 | -5.71116E+18 |
| 843 | -6.01175E+18 |
| 844 | -6.32816E+18 |
| 845 | -6.66122E+18 |
| 846 | -7.01181E+18 |
| 847 | -7.38085E+18 |
| 848 | -7.76931E+18 |
| 849 | -8.17823E+18 |
| 850 | -8.60866E+18 |
| 851 | -9.06175E+18 |
| 852 | -9.53868E+18 |
| 853 | -1.00407E+19 |
| 854 | -1.05692E+19 |
| 855 | -1.11254E+19 |
| 856 | -1.1711E+19 |
| 857 | -1.23274E+19 |
| 858 | -1.29762E+19 |
| 859 | -1.36591E+19 |
| 860 | -1.4378E+19 |
| 861 | -1.51348E+19 |
| 862 | -1.59313E+19 |
| 863 | -1.67698E+19 |
| 864 | -1.76525E+19 |
| 865 | -1.85815E+19 |
| 866 | -1.95595E+19 |
| 867 | -2.0589E+19 |
| 868 | -2.16726E+19 |
| 869 | -2.28132E+19 |
| 870 | -2.40139E+19 |
| 871 | -2.52778E+19 |
| 872 | -2.66082E+19 |
| 873 | -2.80087E+19 |
| 874 | -2.94828E+19 |
| 875 | -3.10345E+19 |
| 876 | -3.26679E+19 |
| 877 | -3.43873E+19 |
| 878 | -3.61972E+19 |
| 879 | -3.81023E+19 |
| 880 | -4.01077E+19 |
| 881 | -4.22186E+19 |
| 882 | -4.44406E+19 |
| 883 | -4.67796E+19 |
| 884 | -4.92417E+19 |
| 885 | -5.18334E+19 |
| 886 | -5.45614E+19 |
| 887 | -5.74331E+19 |
| 888 | -6.04559E+19 |
| 889 | -6.36378E+19 |
| 890 | -6.69871E+19 |
| 891 | -7.05128E+19 |
| 892 | -7.4224E+19 |
| 893 | -7.81305E+19 |
| 894 | -8.22426E+19 |
| 895 | -8.65712E+19 |
| 896 | -9.11275E+19 |
| 897 | -9.59237E+19 |
| 898 | -1.00972E+20 |
| 899 | -1.06287E+20 |
| 900 | -1.11881E+20 |
| 901 | -1.17769E+20 |
| 902 | -1.23968E+20 |
| 903 | -1.30492E+20 |
| 904 | -1.3736E+20 |
| 905 | -1.4459E+20 |
| 906 | -1.522E+20 |
| 907 | -1.6021E+20 |
| 908 | -1.68642E+20 |
| 909 | -1.77518E+20 |
| 910 | -1.86861E+20 |
| 911 | -1.96696E+20 |
| 912 | -2.07048E+20 |
| 913 | -2.17946E+20 |
| 914 | -2.29417E+20 |
| 915 | -2.41491E+20 |
| 916 | -2.54201E+20 |
| 917 | -2.6758E+20 |
| 918 | -2.81663E+20 |
| 919 | -2.96488E+20 |
| 920 | -3.12092E+20 |
| 921 | -3.28518E+20 |
| 922 | -3.45809E+20 |
| 923 | -3.64009E+20 |
| 924 | -3.83168E+20 |
| 925 | -4.03334E+20 |
| 926 | -4.24562E+20 |
| 927 | -4.46908E+20 |
| 928 | -4.70429E+20 |
| 929 | -4.95189E+20 |
| 930 | -5.21251E+20 |
| 931 | -5.48685E+20 |
| 932 | -5.77564E+20 |
| 933 | -6.07962E+20 |
| 934 | -6.3996E+20 |
| 935 | -6.73642E+20 |
| 936 | -7.09097E+20 |
| 937 | -7.46418E+20 |
| 938 | -7.85703E+20 |
| 939 | -8.27055E+20 |
| 940 | -8.70585E+20 |
| 941 | -9.16405E+20 |
| 942 | -9.64637E+20 |
| 943 | -1.01541E+21 |
| 944 | -1.06885E+21 |
| 945 | -1.1251E+21 |
| 946 | -1.18432E+21 |
| 947 | -1.24665E+21 |
| 948 | -1.31227E+21 |
| 949 | -1.38133E+21 |
| 950 | -1.45404E+21 |
| 951 | -1.53056E+21 |
| 952 | -1.61112E+21 |
| 953 | -1.69592E+21 |
| 954 | -1.78517E+21 |
| 955 | -1.87913E+21 |
| 956 | -1.97803E+21 |
| 957 | -2.08214E+21 |
| 958 | -2.19173E+21 |
| 959 | -2.30708E+21 |
| 960 | -2.4285E+21 |
| 961 | -2.55632E+21 |
| 962 | -2.69086E+21 |
| 963 | -2.83249E+21 |
| 964 | -2.98157E+21 |
| 965 | -3.13849E+21 |
| 966 | -3.30367E+21 |
| 967 | -3.47755E+21 |
| 968 | -3.66058E+21 |
| 969 | -3.85324E+21 |
| 970 | -4.05605E+21 |
| 971 | -4.26952E+21 |
| 972 | -4.49423E+21 |
| 973 | -4.73077E+21 |
| 974 | -4.97976E+21 |
| 975 | -5.24185E+21 |
| 976 | -5.51774E+21 |
| 977 | -5.80815E+21 |
| 978 | -6.11384E+21 |
| 979 | -6.43562E+21 |
| 980 | -6.77434E+21 |
| 981 | -7.13088E+21 |
| 982 | -7.50619E+21 |
| 983 | -7.90125E+21 |
| 984 | -8.31711E+21 |
| 985 | -8.75485E+21 |
| 986 | -9.21563E+21 |
| 987 | -9.70067E+21 |
| 988 | -1.02112E+22 |
| 989 | -1.07487E+22 |
| 990 | -1.13144E+22 |
| 991 | -1.19099E+22 |
| 992 | -1.25367E+22 |
| 993 | -1.31965E+22 |
| 994 | -1.38911E+22 |
| 995 | -1.46222E+22 |
| 996 | -1.53918E+22 |
| 997 | -1.62019E+22 |
| 998 | -1.70546E+22 |
| 999 | -1.79522E+22 |
| 1000 | -1.88971E+22 |
Now, perhaps I did the math wrong....but I don't think so.
If ya wanna say something like, "Martingale wins over an infinite time with infinite bankroll etc. as long as there is a winning spin [and you quit after the winning spin]" then go ahead, but that's just as stupid as saying, "My system wins if I win!" or "Ya can't lose if you walk away a winner!"
BTW: EV is additive. All you have to do to figure out your total (net) EV is to add all the EV you've generated (or plan to generate going forward) together. The EV on roulette is -5.26% or -0.0526 of the wager made. Every time you make a wager in roulette, multiply it by -0.0526. Do whatever the hell ya wanna do, increase the bets, decrease the bets, or make no bets at all. No matter what, the EV will NOT be a positive number. The greatest it can be is 0 (ie: you don't make any wagers).
EDIT: For the EV, I'm getting [1 - (20/38)^INF] - [(20/38)^INF * (2^INF -1)]
Which appears to be 1 - INF
EDIT EDIT: https://www.wolframalpha.com/input/?i=lim+x-%3EINF+:+%5B1+-+(20%2F38)%5Ex%5D+-+%5B(20%2F38)%5Ex+*+(2%5Ex+-1)%5D
You gotta copy/paste it.
Quote: DiscreteMaths2Quote: PeeMcGee
Let W be a discrete random variable (winnings) taking values w0, w1, … with probabilities p0, p1, … respectively. Then the expected value of this random variable is:
Because you are leaving out where you lose money. First spin is 1*(18/38) - 1(20/38) not just 1*(18/38).
You are misunderstanding this a bit. The calculation is the sum from n=1 to infinity of (winnings for a series terminating on spin n)*(probability of ending on spin n)
Quote: IbeatyouracesQuote: PeeMcGee...It can be difficult to grasp…but it is in fact zero. In fact it is…
I'll say this again. You are incorrect. If something has a probability of even 0.0000(insert 5,000,000 quadrillion quadrillion quadrillion more zeros)000001% chance of happening, it's still NOT zero.
That expression is equal to zero. If you think it is not zero, what do you think it is?
PeeMcGee posted a claim the EV is 1 in which he defined the random variable and the probabilities and then calculated the expected value of that variable. I think he is correct. For those of you saying he is wrong, where is the flaw in his argument?
My only question is if the probabilities can be defined the way we have done it. I don't know if the probability density function is problematic in some way.
Prob(win) * payout - Prob(loss) * loss_amount = EV
I see no prob(L)*Loss_Amount in the formulas that "show" the EV equals 1.
https://www.youtube.com/watch?v=Nfc7G4xKsl8
Quote: AxelWolfSo is the reason people want to make the argument that infinite time and money using martingale or whatever will work is so they can somehow claim that it can work with less time and money as well?
No basically it was said in passing that a betting system would have positive EV if they had infinite bankroll, limits, time, etc. I challenged that and said no, not even under those conditions would it have positive EV. I am pretty sure everyone is in agreement that this no bearing in real world gambling application, we are just discussing what it means if it could theoretically happen.
Quote: MrGoldenSun
You are misunderstanding this a bit. The calculation is the sum from n=1 to infinity of (winnings for a series terminating on spin n)*(probability of ending on spin n)
PeeMcGee posted a claim the EV is 1 in which he defined the random variable and the probabilities and then calculated the expected value of that variable. I think he is correct. For those of you saying he is wrong, where is the flaw in his argument?
My only question is if the probabilities can be defined the way we have done it. I don't know if the probability density function is problematic in some way.
Expected value for a discrete random variable is a weighted average of ALL possible outcomes. You don't get to ignore certain outcomes and still call it EV.
Quote: DiscreteMaths2No basically it was said in passing that a betting system would have positive EV if they had infinite bankroll, limits, time, etc. I challenged that and said no, not even under those conditions would it have positive EV. I am pretty sure everyone is in agreement that this no bearing in real world gambling application, we are just discussing what it means if it could theoretically happen.
This is accurate. No one here thinks the martingale is positive expectation in the real world. It's a pure thought exercise.
Quote: DiscreteMaths2Expected value for a discrete random variable is a weighted average of ALL possible outcomes. You don't get to ignore certain outcomes and still call it EV.
Okay, what outcome do you think has been ignored?
Quote: IbeatyouracesQuote: PeeMcGee...It can be difficult to grasp…but it is in fact zero. In fact it is…
I'll say this again. You are incorrect. If something has a probability of even 0.0000(insert 5,000,000 quadrillion quadrillion quadrillion more zeros)000001% chance of happening, it's still NOT zero.
We are going to keep going back and forth on this …but that is zero. The crap in the brackets is equal to 0.999 repeating—which is equal to 1 (there are plenty of proofs out there if interested).
Quote: RSTo me, it appears the EV only decreases with the more wagers you make.
Yes, very true. I said that a few pages ago. The infinite Martingale does not work for any finite number of bets. But something interesting happens when you can make infinitely many bets—the probability of going bankrupt becomes exactly 0. You literally cannot lose. Which is why you don’t see it in the expected value calculation.
Quote: RSIf ya wanna say something like, "Martingale wins over an infinite time with infinite bankroll etc. as long as there is a winning spin [and you quit after the winning spin]" then go ahead, but that's just as stupid as saying, "My system wins if I win!" or "Ya can't lose if you walk away a winner!"
So yea, it is kind of like saying that. I would maybe phrase it “The system wins, because it cannot lose”. And it is kind of a silly thing to say—which is kind of how I meant it to be when I first said it. But there are concepts and math behind it that are extremely important. I had no idea it was going to be this controversial of an idea. Now, I’m just mind blown at the unfamiliarity towards these concepts.
Quote:Quote: DiscreteMaths2Because you are leaving out where you lose money. First spin is 1*(18/38) - 1(20/38) not just 1*(18/38).
Quote: RSI don't know what the exact definition of EV is, but I would say it's something like this:
Prob(win) * payout - Prob(loss) * loss_amount = EV
I see no prob(L)*Loss_Amount in the formulas that "show" the EV equals 1.
Quote: DiscreteMaths2Expected value for a discrete random variable is a weighted average of ALL possible outcomes. You don't get to ignore certain outcomes and still call it EV.
I touched on this, but let me add more clarity maybe. My calculation is based on a single round of Martingale. Normally, a Martingale round terminates in one of two ways: (1)by a win or (2)by bankruptcy of the bettor.
So you would see ev calculated like
(1-P[of going bankrupt])*1 – P[of going bankrupt]*bankroll
And that is your Prob(win)*payout – Prob(loss)*loss_amount that you so badly want to see.
But as mentioned, the probability of going bankrupt is zero with infinite money. I have shown this mathematically and tried to explained it logically. I mean, really, it just makes sense. If you don’t believe that the probability is zero…then please explain how one would go bankrupt with infinite money.
So with infinite money and time, the round only terminates on a winning spin. So what you see in my calculation is the probability that the round will terminate on exactly spin n (this is pn) multiplied by the amount of money won or loss on spin n (this is wn). Then all of the possible products are added together.
Yes, my math accounts for ALL possible outcomes. If you add up all of the pn's you get 1. In other words, if you add up all of the probabilities of terminating on exactly spin n for every single n…you get 1. The probability space is covered. There are no outcomes being ignored.
Quote: AxelWolfSo is the reason people want to make the argument that infinite time and money using martingale or whatever will work is so they can somehow claim that it can work with less time and money as well?
I can promise you that is nowhere near my motive. I will reiterate, this works if and only if one has infinite time and money. And like RS have shown, the closer you get to infinite time and money the more you are going to lose. I visit these forums as a mathematician, not a gambler.
I was not referring to you specifically, hell I don't even know who brought it up this time. Others(good movie BTW) have mentioned this in the past and I suspect they were using it as a pro argument for betting systems.Quote: PeeMcGeeQuote: IbeatyouracesQuote: PeeMcGee...It can be difficult to grasp…but it is in fact zero. In fact it is…
I'll say this again. You are incorrect. If something has a probability of even 0.0000(insert 5,000,000 quadrillion quadrillion quadrillion more zeros)000001% chance of happening, it's still NOT zero.
We are going to keep going back and forth on this …but that is zero. The crap in the brackets is equal to 0.999 repeating—which is equal to 1 (there are plenty of proofs out there if interested).
Yes, very true. I said that a few pages ago. The infinite Martingale does not work for any finite number of bets. But something interesting happens when you can make infinitely many bets—the probability of going bankrupt becomes exactly 0. You literally cannot lose. Which is why you don’t see it in the expected value calculation.Quote: RSIf ya wanna say something like, "Martingale wins over an infinite time with infinite bankroll etc. as long as there is a winning spin [and you quit after the winning spin]" then go ahead, but that's just as stupid as saying, "My system wins if I win!" or "Ya can't lose if you walk away a winner!"
So yea, it is kind of like saying that. I would maybe phrase it “The system wins, because it cannot lose”. And it is kind of a silly thing to say—which is kind of how I meant it to be when I first said it. But there are concepts and math behind it that are extremely important. I had no idea it was going to be this controversial of an idea. Now, I’m just mind blown at the ignorance towards these concepts.
I touched on this, but let me add more clarity maybe. My calculation is based on a single round of Martingale. Normally, a Martingale round terminates in one of two ways: (1)by a win or (2)by bankruptcy of the bettor.
So you would see ev calculated like
(1-P[of going bankrupt])*1 – P[of going bankrupt]*bankroll
And that is your Prob(win)*payout – Prob(loss)*loss_amount that you so badly want to see.
But as mentioned, the probability of going bankrupt is zero with infinite money. I have shown this mathematically and tried to explained it logically. I mean, really, it just makes sense. If you don’t believe that the probability is zero…then please explain how one would go bankrupt with infinite money.
So with infinite money and time, the round only terminates on a winning spin. So what you see in my calculation is the probability that the round will terminate on exactly spin n (this is pn) multiplied by the amount of money won or loss on spin n (this is wn). Then all of the possible products are added together.
Yes, my math accounts for ALL possible outcomes. If you add up all of the pn's you get 1. In other words, if you add up all of the probabilities of terminating on exactly spin n for every single n…you get 1. There are no outcomes being ignored.
I can promise you that is nowhere near my motive. I will reiterate, this works if and only if one has infinite time and money. And like RS have shown, the closer you get to infinite time and money the more you are going to lose. I visit these forums as a mathematician, not a gambler.
-------------------------------------------------------------------------------------
I think it would be interesting to follow around and document a martingaler or system bettor for a few weeks.
People are strange.Quote: DiscreteMaths2. I am pretty sure everyone is in agreement that this no bearing in real world gambling application, we are just discussing what it means if it could theoretically happen.
Some people are strange AND disagreable ;-)Quote: RSI disagree about using P(L) = 0 for this equation. P(L) = (20/38)^N for N rounds in a trial.
Quote: RSI disagree about using P(L) = 0 for this equation. P(L) = (20/38)^N for N rounds in a trial.
Please explain how it is possible to deplete an infinite bankroll?
Quote: PeeMcGeePlease explain how it is possible to deplete an infinite bankroll?
Please explain how it is possible to HAVE an infinite bankroll?
or maybe
Please explain how it is possible to increase an infinite bankroll by 1?
An infinite bankroll cannot exist, so this whole thread has been led down a blind alley.
Also PMG.... When you set off with your infinite bankroll intent on Martying until you win, HOW do you know you will win. Your infinite series seems to suggest that you KNOW you will win, but you might be infinitely unlucky and not win even after all eternity has passed. What's ev then?
Quote: OnceDearPlease explain how it is possible to HAVE an infinite bankroll?
I have said it so many times now—infinite bankroll and time is not possible. But that’s not the question we are asking. This is a hypothetical exercise.
Quote: OnceDearor maybe
Please explain how it is possible to increase an infinite bankroll by 1?
I have addressed this multiple times as well. The purchasing power of the bankroll has no influence on the expected value of the game.
Quote: OnceDearAn infinite bankroll cannot exist, so this whole thread has been led down a blind alley.
Also PMG.... When you set off with your infinite bankroll intent on Martying until you win, HOW do you know you will win. Your infinite series seems to suggest that you KNOW you will win, but you might be infinitely unlucky and not win even after all eternity has passed. What's ev then?
Well first, assuming infinite time means eternity will not ‘has passed’.
Second, that would simply mean the bet is still in progress/not resolved.
When calculating the expected value of a normal roulette spin, do you include the time when the wheel is spinning?
Quote: PeeMcGeePlease explain how it is possible to deplete an infinite bankroll?
Seriously? You're twisting all this around. Probability of a loss is (20/38)^INF. This is NOT 0, but approaches 0. Use this in the formula I gave before. Overall, the EV is -INF (big surprise, matches with theory, of -5.26%). You're taking numbers that approach 0 or 1 and then saying they are 0 or 1 when there's more math to be done.
Quote: PeeMcGeeYes, very true. I said that a few pages ago. The infinite Martingale does not work for any finite number of bets. But something interesting happens when you can make infinitely many bets—the probability of going bankrupt becomes exactly 0.
With an infinite bankroll and a finite number of bets, the probability of bankruptcy is also zero. So it really isn't all that interesting.
What is interesting is that everyone agrees you can't have an infinite bankroll, but still allow themselves to pretend or imagine one; yet we can't also imagine losing an infinite amount of times in a row? If you play for in infinite amount of time you will have an infinite amount of losses (along with an infinite amount of wins), why can't they all be consecutive at some point?
What's really interesting is that EV of each bet is -0.027% but EV of a finite series is 1.00? (Ending after a win is finite, not infinite). Even google doesn't give much help in answering these paradoxes.
Quote: RSSeriously? You're twisting all this around. Probability of a loss is (20/38)^INF. This is NOT 0, but approaches 0. Use this in the formula I gave before. Overall, the EV is -INF (big surprise, matches with theory, of -5.26%). You're taking numbers that approach 0 or 1 and then saying they are 0 or 1 when there's more math to be done.
The limit of the total expected value, as the size of the bankroll increases, diverges to negative infinity. You show this with your 1000 row table. No disagreement there.
Your mistake is thinking 'infinite bankroll' simply means 'really big bankroll'. You are still using infinity as a number—while it is more of a conceptual idea. You, and many others, are completely ignoring what it means to have an infinite bankroll.
Infinite bankroll simply means you cannot run out money.
When you have such a bankroll, there is a discontinuity in the expected value as it is equal to +1. Limits do not have to equal the value of the function.
So again, I ask, please explain how it is possible to deplete an infinite bankroll?
Quote: TomGWith an infinite bankroll and a finite number of bets, the probability of bankruptcy is also zero. So it really isn't all that interesting.
What is interesting is that everyone agrees you can't have an infinite bankroll, but still allow themselves to pretend or imagine one; yet we can't also imagine losing an infinite amount of times in a row? If you play for in infinite amount of time you will have an infinite amount of losses (along with an infinite amount of wins), why can't they all be consecutive at some point?
What's really interesting is that EV of each bet is -0.027% but EV of a finite series is 1.00? (Ending after a win is finite, not infinite). Even google doesn't give much help in answering these paradoxes.
I have no qualms with saying that the wheel can infinitely spin one way. That would not deplete the bankroll though. The bet will still be in progress/forever ongoing…meaning no outcome to the bet. Hence, mathematically, this has exactly zero probability.
Quote: PeeMcGeeThe limit of the total expected value, as the size of the bankroll increases, diverges to negative infinity. You show this with your 1000 row table. No disagreement there.
Your mistake is thinking 'infinite bankroll' simply means 'really big bankroll'. You are still using infinity as a number—while it is more of a conceptual idea. You, and many others, are completely ignoring what it means to have an infinite bankroll.
Infinite bankroll simply means you cannot run out money.
When you have such a bankroll, there is a discontinuity in the expected value as it is equal to +1. Limits do not have to equal the value of the function.
So again, I ask, please explain how it is possible to deplete an infinite bankroll?
I have no qualms with saying that the wheel can infinitely spin one way. That would not deplete the bankroll though. The bet will still be in progress/forever ongoing…meaning no outcome to the bet. Hence, mathematically, this has exactly zero probability.
I never said or meant to say the BR could be depleted. I'm saying the EV is negative. I could play a -EV game (9/6 JOB straight up) with an infinite bankroll, and my bankroll would not be depleted. But, the game is still -EV.
Quote: RSQuote: PeeMcGeeThe limit of the total expected value, as the size of the bankroll increases, diverges to negative infinity. You show this with your 1000 row table. No disagreement there.
Your mistake is thinking 'infinite bankroll' simply means 'really big bankroll'. You are still using infinity as a number—while it is more of a conceptual idea. You, and many others, are completely ignoring what it means to have an infinite bankroll.
Infinite bankroll simply means you cannot run out money.
When you have such a bankroll, there is a discontinuity in the expected value as it is equal to +1. Limits do not have to equal the value of the function.
So again, I ask, please explain how it is possible to deplete an infinite bankroll?
I have no qualms with saying that the wheel can infinitely spin one way. That would not deplete the bankroll though. The bet will still be in progress/forever ongoing…meaning no outcome to the bet. Hence, mathematically, this has exactly zero probability.
I never said or meant to say the BR could be depleted. I'm saying the EV is negative. I could play a -EV game (9/6 JOB straight up) with an infinite bankroll, and my bankroll would not be depleted. But, the game is still -EV.
The difference is, such a game will not guarantee a profit when you win.
