I worded that awfully. By “small” I obscurely meant “small number of distinct locations on the board”. Better would be: “At a roulette table in that scenario, it is indeed better to place a bet on a single number rather than spread out over multiple numbers”. More generally (though worded only roughly), if your expected earnings are negative, you need Variance, Baby! Variance! if your goal is to maximize the probability of reaching some target (positive) earnings.
Yes! And as I mentioned in an earlier post, if your expected earnings are positive (e.g. playing the stock market or operating a casino) then you want to avoid Variance (i.e. you should hedge your bets).
Before further confusion arises, let me summarize how you should really bet next time you’re in a real casino! Let’s assume you “plan to lose” $500.
Most people gamble for fun. You make relatively small bets, hoping that that $500 lasts the whole weekend. Whether you bet $2 or $20 at a turn will depend on how much you need to bet to get the “thrill” you want, i.e. what achieves the entertainment for which you’re there at all.
But let’s assume that you don’t seek many hours of entertainment; that instead you really do want to maximize your chance of going home with large winnings, even if that strategy means it’s likely you’ll lose your whole $500 bankroll in a few minutes.
Go to the baccarat table and bet the whole $500 at once on an even-money bet! This is not a high-payoff bet, but it is still a “bold play,” to use Lance Turbo’s term. If you win … let it all ride! And let it all ride again until you reach $4000, or whatever target you’ve set. This may be a very bad strategy if your real goal is to drag the gambling entertainment out over the whole weekend, but it is a very good strategy to maximize your chance of going home with $4000.
The reason I suggest an even-money baccarat bet rather than a 35-1 roulette bet is that the vigorish at American roulette is much larger than the vigorish at baccarat – so much larger that that disadvantage more than compensates for the advantage of the higher payoff. (Blackjack and craps also offer very low vigorish, but detailing the best approach would be complicated by the need to keep part of your bankroll in reserve for doubling-down, odds-taking, etc.)
If you refuse to play anything but roulette and want to go home with $18,000, your best strategy (if the casino will permit it) is to bet the whole $500 at once on a single number. If, instead, you’ll be satisfied to go home with just $4000 your best strategy is to bet $100 on a single number and quit if it wins. If it loses bet $103 on a single number, followed by $106 after another loss, and so on.
I’m making this post to demonstrate that the principles that Lance Turbo and I enunciate in this thread are not an abstract solution to an abstract logic problem, but reflect your best strategy in Las Vegas if your goal is to go home as a big winner. (Yes, yes, yes; in practice the entertainment aspect of casino play is so important that these strategies – where you usually lose your entire bankroll quickly – won’t make sense for you.)
And again, they apply when your bets have negative expectation. If you’re counting cards at Blackjack, or “betting” in the stock market, the principles are quite different.
I get that. That is why I keep quoting him and asking him to defend the statement. He keeps changing the scenario. The initial issue I had was him saying:
So if he meant that such a strategy was superior in the limited circumstances of trying to double your money or whatever, he could clearly state that rather than arguing about this point.
Again, you keep moving the goalpost. Do you defend your post I quoted previously? Not as a matter of trying to play a limited number of hands to reach some predestined number, but as a broad strategy?
Nice humblebrag!
That is not what we are discussing as I’ve attempted to make clear at least 3 times.
Who agreed with you?
Perhaps we disagree on the notion of “limited circumstances.” IMO, if one is gambling deliberately in hopes of making money it is rational to have a goal. If you want to increase, for example $1000 to a large amount, e.g. $10,000 it is best to make single-number bets. If you want to increase your $1000 to just $1100 then (assuming the casino allows you to bet odd amounts like $2.86) it is best to make single-number bets. If a spectrum of outcomes is acceptable to you (e.g. you’d be happy turning your $1000 into either $2000 or $3000) then … it is still best to make single-number bets.
But in what sense (other than entertainment) is it meaningful not to have a target? Does “not having a target” mean that you haven’t decided what you’ll do if you win a bet? :dubious: Anyway, I’m here to discuss the mathematics of gambling, not English usage. If you consider the three very different cases I’ve mentioned in the previous paragraph to add up to “limited circumstances” then we’ll just agree that we use English language differently. Fine. You win.
In any event I’ve explained over and over very explicitly the claim that I’m making. (And I’ve still not heard explicitly from you whether you accept the truth of that claim.) You seem to think that I phrased an earlier sentence poorly in comparison with my actual claim. I don’t believe that’s true, but suppose it is. Then you’re not arguing with me about the mathematics of gambling. Instead you’re chortling over a Gotcha! “Septimus may be correct about his main point, but he worded his earlier claim poorly. Ha ha ha! Gotcha!” I don’t wish to play that game.
In any event, this conversation reminds me too much of the earlier exasperating thread. I hope not to post again in this thread. Let’s just declare brickbacon the victor, OK?
Do you think most people gamble to make a specific amount of money in profit ? Of course not. They may gamble with the vague hope of making money, but not a specific amount in the majority of cases. In fact, most people realize they aren’t likely to make any money at all.
Most gamblers do not have a target.
You would have a point if I were the only one who misunderstood you. I wasn’t, and you subsequently doubled down on the misunderstanding despite me quoting you and asking for clarification . If there is anyone at fault, it’s you.
Thanks all for the helpful replies.
The window dressing of this random draw as being a poker tournament may be relevant in a way. The pay structure is handled exactly like a ‘real’ poker tournament in that the top 10% of the field get paid something, and the amount goes up the longer you survive.
The reason this seemed relevant to my original question is that in a ‘real’ poker tournament one can play to win first place, but one can also play just to survive long enough to be in the money. In some cases in huge tournaments you can literally fold every single hand until all remaining players are in the money, guaranteeing yourself at least a small payout. Then you can start playing boldly to last longer or even make first place from way behind, knowing that if you get knocked out you are already guaranteed a small payout.
When a tournament is on the ‘bubble’, e.g. only one player needs to be knocked out before everyone left is guaranteed a payout, players will fold incredible starting hands like AA, KK, etc. just because they know if they get knocked out that hand they will get nothing, but if they survive just one more hand they get something. So they will fold a hand that by all long term odds they definitely should play.
Poker pros will tell you that you should always play to win, not just to survive until the money. If you get knocked out on the bubble, so be it. It was the right play to make and in the long run it was the winning play to make. But they are talking about the long term odds over a long time. In the OP I did ask about the long term odds but realistically we don’t have infinite time to work with. At best we have my lifetime and more likely we have only until I get tired of messing with it and stop entering, or the poker room stops the promotion. So without 2 trillion iterations to wait for expected return to even itself out, it seemed like doubling my chances in any one tournament may be better in a short term, “at least I got something”, way.
But from the many helpful replies here, it looks like even if there were only ever to be two tournaments, and I had to chose to either play each one with one entry or only one with two entries, the chance to walk away with ‘at least something’ is still the same.
Yes, this is for sure the best strategy. But it is unfortunately hard to predict as even in the last minute leading up to the tournament thousands of entries may pour in last minute, from players using the same strategy. One of the benefits of this tournament is you don’t even need to be online to play, so if I had to actually sit there watching the number of entries at tournament time and making last minute decisions about whether to enter or not, the hassle factor may outweigh any improvement to my odds.
Maybe we’re in closer agreement than you think! I certainly did not deny the reality of recreational gambling:
As I say, I do review my posts for clarity before clicking Submit. I did see that my affirmation of recreational gambling was emphasized to excess in this post, but I thought it might be best to be very clear about this important practical point.
Apparently though, my recognition of the reality of recreational gambling was not excessive or redundant enough! 
Say the goal is “get the maximum amount of $X bets before I halve my bankroll.” On average, what’s the best betting pattern?
I’ll assume your wager X=1, and your bankroll is 2BX. (Just multiply any dollar amounts by X.)
First of all: Play a low-vigorish game, i.e. blackjack, baccarat, or betting the Line or Do_Not at craps. However, since one of the themes of this thread is high-payoff versus low-payoff bets, I’ll assume instead that you play roulette where the high-payoff bet is available.
Second: Always make your minimum bet. Any increase will tend to lead to ruin sooner. An exception is that you take certain permitted bet increases, i.e. Odds at craps, or splitting Eights and Aces and Doubling Down on 11, etc. at blackjack.
Third: Your question is somewhat ill-formed. It’s easy to calculate the average number of bets you’ll get before ruin, but there is no upper limit on how many bets you might get to make.
Now the answers!
The average number of $1 wagers you will make at American roulette before halving your bankroll 2B is *precisely 19B no matter how you bet! (This is such an elegant result, it seems best to leave it as exercise to encourage others to participate in the thread. 
)
But what about the variation in results? If B=10 and you bet on Red, you’ll have about 2% chance of being able to make 1000 or more bets, and about 27.5% chance of being able to make 200 or more bets. If instead you place single-number bets, your chance of making at least 1000 bets increases to 3%, but your chance of making at least 200 bets falls to 8%.
The percentages in the preceding paragraph were derived by a simulater I threw together. The simulator did each experiment 1,000,000 times and continued play until you’d lost the entire $10 budgeted.
In the million trials with single-number bets, it once took about 140,000 wagers for you to lose the stake; you’d get as high as $3500 during the million trials. (I write fuzzy numbers here because I actually did the million-trial experiments multiple times.) With even-money bets (e.g. Red) the longest you went before ruin in a million trials was about 6500 bets, and the furthest you ever got ahead was about $150.
On the other hand, with single-number bets and B=10 there is a 76.5% chance you only get to make 10 bets, while betting Red there is a 82.4% chance you get to make 40 bets or more.
So if the question was formulated as “a player has A hours to play. If they make B bets per hour, how can they maximize their likelihood of playing all A hours?” The answer would be that they make small low variance bets. That is it would be better to bet on red than on a single number. Correct?
Yes, as long as AB isn’t very large. But if AB is sufficiently large then the situation reverses at American roulette and the high-payoff bet becomes better: as the results of my simulation showed, $10 has a 3% chance of surviving A*B = 1000, compared with just 2% for the even-money bet. (But that’s comparing with the even-money roulette bet. You’d always be much better off at one of the low-vigorish games: baccarat, craps or blackjack.)
Maximizing time to ruin rather than chance for a big win may strike some as odd! Why not just factor recreational value into the dollar amounts?
I’m afraid focus on variance may lead to confusion; I’d describe a bet more “purely” by vigorish and payoff. To complete part of the discussion, let’s comparing roulette’s 5.26% vigorish 35-1 bet with an even-money bet. If the goal is to increase your bankroll 10% then roulette roughly equals the performance of an even-money bet with 3.43% vigorish. But if the goal is to increase your bankroll 1000-fold, the even-money bet must have only 1.25% vigorish to equal roulette. (And a 1000-to-1 Keno ticket would need 12% vigorish to equal the roulette play; but Keno vigorish is about 30% IIRC.)
Hmm, I think it’s more in line with how the casual gambler operates. I think visitors in Vegas probably go hit the slots or whatever in the 2 hours between dinner and a show without a single thought about a goal of doubling their money or any other specific dollar amount. If we’re talking utility I think it’s highly non-linear and discontinuous at both the “gone broke” point and the “broke even” point. Other than that I don’t think it usually matters.
I’m not sure though. I don’t have that inclination. I only play poker where I can have a, potentially misguided, positive expected value.
Another thought about the maximally efficient way of winning a certain dollar amount: say there’s an individual that is playing this strategy on accident or is interrupted or is in some other way not able to carry it out to conclusion. That player is more likely to leave the table down money than the “naïve” player that’s betting on red.
I think talking about variance is instructive since imagining the guassians of low vs high variance bets gives a good indication that there is “crossover point” in which one strategy will give you a higher likelihood of being in the black than the other.
In the earlier simulation with effective bankroll $10, you make 190 bets on average, going home with $Zero 100% of the time. (It takes 150,000 wagers to ruin in some cases, affecting the statistics significantly.)
A more reasonable termination condition would be
Bet $1 on either Red or single-number until EITHER (a) you’ve lost $10, OR (b) you’ve made 1000 bets.
I did simulate this; the results might be interesting.
Betting on Red, you now make 181 bets on average, going home with $Zero 98% of the time, but (on average) $23.60 2.05% of the time, for an average loss of $9.52.
Betting on single-number, you make 73 bets on average, going home with $Zero 97% of the time, but (on average) $209.00 2.95% of the time, for average loss of $3.82.
So, with these conditions, your average loss is reduced, you get more winning sessions (3% instead of 2%) and win more in those sessions ($209 average instead of $24). I think this may make the single-number betting look better, even for the entertainment purpose. Note that the $209 wins become $10,500 wins if your bank is ten $50 bets instead of ten $1 bets. (Of course you still have the problem that with only $10 bank you’re ruined immediately 76.5% of the time).
Let’s try B=$20. Now with single-number you make 210 bets on average, winning an average $210 6.0% of the time for a net average loss of $7.40. (Immediate ruin is 58.64%.)
With Red bet, you make 351 bets on average, winning $24.3 6.2% of the time, for a net average loss of $18.48.
I’d be happy to repeat the simulation for parameters other than ($10 or $20 bank; 1000 wagers).
The problems are defined via 1st statistical parameter (vigorish) and 2nd statistical parameter (payoff or variance). The 1st moment is the much more important parameter. I may investigate whether
f(Vig, Variance, Goal) = Effective Vig
plots more insightfully than
f(Vig, Payoff, Goal) = Effective Vig