### Martingale Variation

I have a question, if it's not too much of a pain in the backside.

A little context first to clarify my question.

We know that your chance of winning a single bet is 18/38 on an outside bet of even/odd or red/black. But suppose that's not what you are actually hedging on. Rather, you're playing the long game. You sit, you watch, you calculate that missing red 20X in a row = 20/3020, which is basically a phenomenally "rare" event when compared to the total trend. Can you exploit this? Let's suppose a minimum outside bet of \$1 and a max outside of \$100, and you are attempting to stretch your dollars. So rather than double up at every turn, you simply bet in order to re-coop losses if they occur.

You sit at the wheel and notice that red hasn't hit 3X in a row, so you bet on it, because you are hedging your bet that you will not be present to witness red missing 11 times in a row. 47.378421% chance you'll make a dollar on the first chip placement of your betting proces, and about a 52.535668% chance you'll just recoop your loss in the subsequent phases of the betting process, (you're betting in sequence 1, 1, 2, 4, 8, 16, 32, 64 for a total of \$128 you are willing to invest in this experiment) and about 0.00085911% chance you'll eat shit for a total \$128 by playing this martingale all the way through. Think of this all as one giant bet of which you have wagered \$128 against the house.

Probability of winning times the value of a win minus the probability of losing times the value of the bet = expected value. Right?

Your expected value of winning the betting process is 18/38(\$1) = \$0.4736...etc

Your probability of a loss (18/3811) times \$128 = \$0.1099661

Total expected value is 0.4736 - 0.1099 = ~0.3637

First up, my common sense tells me that I must have done the math wrong since I'm in the positive. But I can't quite figure out where. I'm only hedging a bet that I won't be there when a given outside 1:1 payout bet misses 11 times in a row. In other words, I won't bet on the table at all unless those conditions are met. That can be for Red, Black, Even or Odds, doesn't matter. I'm just hedging my bet that I won't be there to see that happen on the first thing I bet on. For example, if I see red has missed 3x, and odds has missed 2x, I bet on red. Now, it may come to pass that the red hits after two more spins, I simply recoop my losses, yet odds has still not hit. At this point, I switch my bet to odds. I again have a 18/38 chance of making a dollar, and I am now betting my \$128 that I will not witness odds miss (2 + 2 + 8) 12x in a row. That's a preferable bet, on the face of it.

Where is my math wrong, and at what point (number of proposed misses) can I say "look, I can reasonably expect it to take a week of non-stop games before I see this outside bet missing this many times in a row?"