sirius said:
What I was trying to explain was that you are just as likely to increase your bankroll by that amount or more playing the slots (most players will have more chance of boosting it a lot playing the slots as they need to bet big in the other games). It is flawed to ban roulette and blackjack. Slots actually have much higher variance so the house edge isn't a significant consideration for many thousands of spins.
The expected loss you calculated for slots assumes you will wager $3000 but, due to the variance, the player loss is more likely to be less than $50 on average because many times the bankroll will be wiped out well before even a third of the requirements are wagered.
This is true.....
If we consider a hypothetical simplified slot with the following (generous) paytable:
Probability Pay
.00002 10,000
.035 10
.4 1
The payout can be seen to sum to 95%, and the WR is $3000 with a $1 stake. Given our initial bank of $50 deposit + $100 bonus = $150, it seems we should lose $3000 *.05 = $150, or lose all our money. So the offer seems to be bad for us.
Based on a 10,000,000 trial simulation, we get an average final balance of $119.65. Ths is a $69.65 profit on something that appears to have no advantage!
Why? Well, the average number of spins was 632.6, because if we hit the jackpot we will definitely do all 3,000 spins without busting, but if not then it's highly unlikely. So the actual practical average WR is $633. This sounds a lot better than $3,000! The downside is the reason for that low WR is that you have lost all your money most of the time....
So using the actual average wagered amount, 632.6 * 0.05 = $31.63 expected loss. So rather than losing $150 (all the bonus + the deposit) on average, we lose far less. If the casino hadn't given us a bonus, then we would have lost $31.63 on average (where in order for the mean of the sampling distribution to appear normal, you need to playing this offer thousands of times at the least), which is as you would expect for a game with a house edge, but as they gave us $100, so we are ahead.
The downside is that we have a 98.8% chance of busting out, so the offer is very high risk to lose $100, much less to win.
This wouldn't work where you have to meet the WR to get the bonus, only when you get the bonus in advance.... Or if you continued to the WR, redepositing (which would be pointless, given that there is no more bonus to get). So the correct strategy that maximizes EV (at the expense of horrible risk to your bankroll) on a given bonus-paid-before-WR-met promotion is to reduce the number of expected number of spins to a minimum by choosing a game that has most of its payout as the jackpot. Obviously a low house advantage is good as well, but given a $3000 WR, and the choice between a 95% return on a low-variance slot with an average of 2,000 spins before busting, or a 90% return on average of 800 spins, the 90% slot has a higher EV (but much higer variance).
Unfortunately my example above is purely theoretical, and I have no idea whether it bears any resemblance to any actual slot..... It's much easier to find probabilities for video poker than for slot machines. But if you had the pay table and probability for a casino's slots, you could compare them all with for the WR and work out the risk of ruin (or rather the risk of NOT getting ruined, which is probably more appropriate when the risk of ruin is 99%!), and decide which is the best one to play.
The odds of hitting a certain target in blackjack and roulette will be slightly higher due to the lower house edge but the max target a player will aim for will not realistically be as high as is possible playing slots. Obviously for all these games, the average result will be negative due to the house edge in all of them.
If we used this hypothetical slot machine, then the average win, excluding bust-outs is $119.65 * 100/1.12 from $150 (now disregarding the bonus) = $9960, or slightly less than the $10,000 (jackpot.
If we played roulette with a $150 balance, with an evens payout on 18/37 odds, betting the whole balance every time, then to get to $10,000 we would need only 6 bets (1 bet taking us to $300, 2 to $600, 3 to $1200, 4 to $2400, 5 to $4800, and 6 to $9600). (18/37)^6 = 0.013.
So in fact a $10,000 target is equally reasonable (1.2% vs 1.3%), even playing a suicidal roulette strategy. It's simply psychology that allows people to feed $150 into a slot machine chasing that improbable win, whereas they are unlikely to keep doubling up from $1200 to get that $9600 win, for fear of losing what they have already got. But in fact in both cases you are jeopardizing your $150 in return for a remote chance of $10,000.