June 14th, 2009
Why is it that so many smart players believe that negative fluctuations are something that only happens to the other guys?
Forget the math for now. Let's look at a practical example of what can happen. In order to demonstrate what you might have to deal with—in the short run, in a very profitable game, even when you're using your system perfectly—I ran 100 consecutive computer simulations of 1,000 hands of blackjack each. Each simulation represents about 10 hours of play in a truly great game: single-deck, Vegas Strip rules, with double after splits, and 80% dealt out. The player is using the Zen Count with all published indices, and spreading his bets from $25 to $100.
His long run advantage in this game is 2.1%, which translates to about $105 per hour. Before I ran the 100 simulations of 1,000 hands each, I ran a 10-million-hand simulation to come up with the 2.1% win rate. Since you probably won't play any 10-million-hand sessions, let's look at what happens in your real world 10-hour sessions. Let's say that you, a dedicated card counter, actually went out on 100 consecutive weekends, and played ten hours of blackjack every weekend for almost two years, in this same game. What happened?
Of these 100 10-hour sessions, you won at least some money on 76 of them, but lost on 24 of them. That sounds great—you lost on less than one out of four 10-hour playing sessions. However, these wins and losses were not evenly distributed. The longest series of consecutive winning sessions not interrupted by a single losing session of any amount, was 18. During this simulated 180 consecutive hours of real-time play, you won an average of $198.50 per hour, almost twice your actual expectation! In dollars and cents, after 180 hours of play in this game, you would expect to be ahead by $18,900.
Due to this fluctuation, however, in this 18-session run you were ahead by $35,640. In your single best playing session, during which you had an expectation of $1,050 after 10 hours of play, you actually won $6,200, which translates to a win rate of 11.76%! Not bad for ten hours of play where you never raised your bet higher than $100…
Hey, if you won an average of 200 bucks an hour, for 180 consecutive hours of play, you'd probably figure it was time to quit your day job! What a marvelous game! Ain't counting wonderful?
But wait a minute. If your wins were fluctuating and clumping so wildly, doesn't that mean your losses did too? You'd better believe it. On your worst losing session, instead of winning your expected $1,050, you lost an even $4,000. And on your worst two consecutive losing sessions, instead of winning your expected $2,100, you lost $7,675. Your worst 10 consecutive sessions contained seven losing sessions and three winning sessions. At the end of this simulated 100 consecutive hours of play, you would expect to be ahead by $10,500; instead, you were behind by $8,175.
And what was the longest series of consecutive sessions in which you still failed to show a net win at the end? Nineteen.
One hundred and ninety hours of consecutive play.
The series contained 11 losing and 8 winning sessions. Your expectation after 190 hours at this game is to be ahead by $19,950. Instead, you were still behind by an even $2,000. This is a negative fluctuation of almost $22,000 below your expectation!
Mercifully, on a game as great as this, there wasn't a series of 20 consecutive sessions in which you did not show at least some small win. So, in this game, if we define "long run" to mean approximately how long you must play before you're more or less assured of being ahead, the answer is about 200 consecutive hours.
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