I was recently given a promotion by an online gambling website for £10 free credit which I could withdraw after having wagered at least £200 in total. What's the best betting strategy for maximising my EV when playing roulette? The non trivial feature of this problem being that, if I lose my initial £10 I cannot continue gambling to achieve the long term expected value of my bets.
I decided at first I was continually betting £0.50 on red. Reasonably quickly I decided to go with a different strategy of betting £0.50 on red and £0.50 black each time meaning I'd break even as long as the ball did not land on 0.
How does one calculate the EV in each scenario, simply betting on a single a colour, or betting on both black and red. It's trivially simple if you have unlimited money, and in both cases the expected value is 36/37~0.973 (since we're playing european roulette with just one green square) however in my case, if I go through a bad streak and lose my £10 I can not continue. Therefore it seems the EV becomes partially dependent on the variance of the strategy.
Secondly what produces a higher EV for both the strategies above, a large amount of small bets, or a smaller amount of larger bets. I.e. one thousand £0.50 bets or one hundred £5?
My gut instinct says that because of this feature that once I'm down a total of £10 I can't continue makes me think that smaller bets are produce a higher expected value but I don't know how to prove it.
Edit: EV stands for expected value, however I may be using the term incorrectly. To explain what I mean here's a simple example of two strategies which are easy to calculate. One where you bet £5 on red and £5 on black every bet, your mean returns is £10*(36/37)^20=~£5.8 i.e. you need to avoid hitting a 0, 20 times.
If instead you bet all the money you have on red every time, you need to win 5 times in a row before you've wagered at least £200, so your mean returns is £320*(18/37)^5=~£8.7 which is greater than £5.8. This is what I meant by expected value. So not the expected value of an individual bet, but the expected value of your winnings after betting at least £200.
Or you could think of it as, if a million people all got the same promotion and then pooled their winnings and shared them out between them, whats the best strategy they should all follow.
Best Answer
The expected return on a total bet of $200$ is $200 \cdot \frac {36}{37}\approx 194.59$ That does not change whether you bet one big lot or many small lots, nor whether you bet high odds things like one number or even money things like red/black. What can change is the chance you are down $10$ at some point in the run. If you make many small bets at even odds the dispersion of results will be reduced, so that is what you should do to have the best chance of getting some money out of the deal.