**How long is the "long run"?**
That’s a great question. Most players know that they are playing for the “long run” but they have no idea how long it takes to get there (or, at least, reasonably close).

You’re right that the outcome of a few hours of play is fairly uncertain (within the limits of SD). In fact, the results after several hundred hours of play are still subject to variance. So how long will we have to play before we have a reasonable chance of reaching our EV? Well, there are two different ways to look at it.

**N0 (N-zero)**
As you know, variance will sometimes be positive and other times be negative. In the long run we expect the “lucky” and “unlucky” times to cancel each other out so that we are left with our EV. The N0 formula will tell us how many hands (or hours) we will have to play in order to overcome the variance. A simplified version of the formula is:

N0 = SD^2/EV^2

So if your hourly EV is 1.5 units and your hourly SD is 28 units you will get:

N0 = 28^2/1.5^2 = 784/2.25 = 348.44

You would have to play for 348 hours (at 100 hands per hour) to overcome one SD (about 68% of the variance). You can expand this to two SDs (about 93%) by multiplying by 4 (the same as 2^2). That would give you about 1,394 hours of play to have a very good chance of reaching your EV. That’s a lot of playing! If you played for 10 hours every week it would take you over 2.5 years to get close to the long run. That’s pretty discouraging for recreational players, but fear not – there are ways to improve the situation.

N0 is basically a comparison of EV to SD. If you can raise your EV, relative to your SD, then you can save yourself some time. By playing more aggressively (Wonging out, backcounting, using a more optimal bet spread, spreading to multiple hands, etc.) you can either increase your EV, reduce your SD, or both! Look what happens when we make a small change to our playing style”

EV=1.8

SD=26

N0 = 676/3.24 = 209

4N0 = 835

Our 4N0 is only 835 hours instead of 1,394. We’ve saved ourselves 559 hours of our lives! That’s over a year that we can spend away from the dark, smoky casinos. Early retirement anyone?

But that’s still over 1.5 years of play. That’s still pretty discouraging, especially if we don’t play that often. Isn’t there a formula for people who only play a few times a year?

**p(ahead)**
Sure there is! Most recreational players do not rely on their BJ winnings for their livelihood. They just want to win a little money on the side. Sure, it can feel disheartening to look back on a year’s worth of play and only see a $100 profit, but it sure beats losing!

For players like this there is a formula for calculating the probability that you will be ahead (by any amount) after a given number of hours of play. This number is often much more encouraging because the probability of being richer, even if only by $1, is much higher than reaching your precise EV.

The formula for this is a bit more complicated. For those interested, you can check out an

old post I made with the details. For those who just want to see the results, read on.

In this case, with an EV of 1.8 units and a SD of 26 units your probability of being ahead is roughly:

Hours___Probability

1________53%

10_______59%

100______75%

500______94%

1000_____99%

Now that’s a lot better, right? You have about a 60% chance of winning any given month. After 1 year of play (10 hours per week, 520 hours per year) you won’t get all the way to the long run, but you will overcome most of the variance and have over a 94% chance of being ahead. It ain’t perfect, but that’s about as much security as you’re going to get in the gambling world.

This is another reason why it is crucial to understand optimal betting strategies. Sometimes a little change in your bet spread or playing style can add up to a huge improvement in your N0. After all, what good is a healthy EV if you never have a chance to get it?

-Sonny-