There is no guarantee of anything. The only thing (almost surely) garanteed is the law of large numbers. Don't be missguided by that mathematical law: It won't say that from a million coin flips you win exactly half a million flips.
It does say that the ratio of all wins and all losses will aproach the chance of that bet.
Again, it doesn't say you are "due" to some result, even on a loosing streak.
A bet being +EV or -EV is independent of short or long runs. You may win or loose in short run, but if your bets are +EV, you are more likely to win in long run.2. Think I probably just need to read the thread over again. I understand why it's +EV to double up in the short run (all the examples are in the short run on page 2, nothing talks about the long run at all).
Yes in that ideal world you would be 790 up if you follow the doubling up strategy. But in that ideal world there are better strategies (triple ups...) which will get your profits nearer the 2k limit (total bonus amount).I think my point is probably valid, just that it's not noticeable. Take a hypothetical world where probability comes true (i.e. if you flip heads, you are "due" a tails next turn). If the odds are taken to be 49.5/50.5 to the house (standard blackjack HE yes?), say, then statistically, you will win at 99 casinos, and lose at 101 for every 200 offers.
Lets say all the casinos are a standard £100, 100%, 40xB (WR=£4000). By doubling up compared to grinding out, you make an extra £10 per offer (see your post 21 on this thread, which I understand).
In the 99 offers you win, and the 99 symmetrical offers you lose, you therefore make an extra £990 compared to grinding. But there are 2 casinos in that 200 where you lose the deposit due to the house edge and don't receive a symmetrical doubling up win. Assuming all the bonuses are the same, you therefore lose £200. So +EV is £790 for doubling up over the 200 casinos.
And yes, following your next point. If you happen to loose on 111 and win on 89, you are down compared to grinding.
The law of large numbers now help you here: if you did 2 million offers, you would be much much more often up (compared to grinding) than down. The reason is that the ratio of wins agsinst losses shrinks (almost surely). And in that position you almost surely make more profit.