Well I was doing some calculations tonight, and figured out something reasonably interesting, but I think it may be wrong somehow.
Let's say you have $1025 in start up capital and you want to use the Martingale betting system at a Las Vegas Blackjack table. Let's say you play reasonably average rules which gives you a 47.49% chance of winning each hand, ignoring ties.
So then, the chance of any consecutive loss is 47.49% X 47.49%. Consequently, the chances of any....say.....10 consecutive loses is 47.49% ^ 10, which just so happens to approximately equal 0.14%.
By that time, you've made 10 dollars. So using Martingale and $1025 in start up capital, you have a 0.9985% chance of making 10 dollars and if you didn't by some cosmic fluke, you'd lose every bit of the $1025.
What about if you wanted to make 100 dollars? That would be 10 SETS of 10 hands, right? Well we know that there is a 99.85% chance we won't lose 10 times in a row on 10 hands, so to calculate risk for 100 dollars gain, we should figure 10 SETS of 10 hands, for a grand total of 100 hands and dollars. That means that the correct probability for not losing everything during our 'run' is 98%, I think.
If THAT's true then it's true that for $1,000 dollars, you'd need to go through 100 sets of 10 blackjack hands (for this $1,000 eventual total profit, but you'd need to play 1,000 blackjack hands over the course of a vacation).
And now when we take .9985 to the 100th power we find that they have a 86.264% chance of pulling something like this off.....right?
Truth is, I'm starting to see one big glaring flaw in my math; can you help me figure out what it is?
Say you're playing at a $5 table with a max bet of $5000. You loose 10 times in a row, now you're betting $5000 (you'd like to bet $5120, but we won't mention this) Loose the 11th time and you're down $5000 even if you win on the 12th time!
The chances of this happening are about 1 in 1000. Say you win $5, on average, every 2 hands for 1000 hands. You've won 500 times after 1000 hands: $2500. Then it hits you, right on schedule: you loose ten in a row. POW! you loose $5000. With your winnings of $2500, you're still out $2500. OUCH!
It's actually worse than this - you don't need to loose 10 in a row if you lose a split or double. You don't know when those 10 in a row are coming - there's more than a 50% chance you'll see 10 losses in a row after only 700 hands. It's hard to find tables with a 1:1000 spread - at a quarter table with a $5000 max, you can only double 7 times. You'll win less than 1 out of 2 hands - more like 1 out of 3. You don't need 10 "losses" in a row: pushes don't count, if you split and win one loose one, it's a push.
Be careful - this method is very deceptive! You'll win 99 hours in a row. Then, when you loose, you loose much more than your total winnings in a single hand. And the House will not give you a "Do Over".