Blackjack Variance

As promised, some math backing up the variance talk in the last blog post. It seems obvious that increasing your bet size will increase your variance, and (amazingly enough) the math supports that! Adding two columns to my previous table for the $100+$100/$2500 WR at 300 hands/hr:


where RoR is your Risk of Ruin (losing your entire $200 bankroll) and RoL is your Risk of Loss (losing more than your $100 bonus, leaving you in the red for the bonus, but not broke). Some of the RoR numbers for the very large bet sizes are likely incorrect due to the model I used, but the $1-$5 bets should be fine.

To perhaps better visualize these numbers, I present the following graph:

• The green line is at $200, your initial bankroll.
  
• The black line is at $187.50, your EV. Note that this is at the center of these curves, and thus your most likely result.
  
• The red line is at $100, the amount of money you deposited.

The way to interpret this graph is that the area under the curve is proportional to your probability of winning that much money.

For example, take the lowest (purple) curve, gotten by flat-betting $1 over the whole bonus. The area of under the curve to the left of the $100 line is your probability of finishing your bonus with under $100. You can see that area is small compared to the entire area under the curve, thus your chance of finishing in the state is also small. From our table above, that "small number" is actually the 6.4% RoL from the $1 line.

Likewise, there is virtually none of the curve that would appear to the left of $0 (none of it looks "chopped off" by the left hand side of the chart, thus your chance of finishing below $0 is vanishingly small. From our table, that chance is 0.1%.

Note several things about this chart:

• Our EV never changes regardless of how much we bet - it is always $187.50. While not strictly correct, it is correct enough for our purposes.
  
• The larger bet sizes have a much larger variance - note for example the amount of the $5 or $10 curve that is to the left of the $100 line, and the amount that looks chopped off. However, this variance is also manifest on the right side of the chart - the only way your will ever end a bonus with $450+ is to bet $10 or more.
  

As you can see, there is some very wide variance, even for very small bet sizes. Even flat-betting $3/hand will have you losing money almost 20% of the time and going broke entirely about 3% of the time, but will also have you making over $30/hr in the long run. I am personally comfortable with this level of risk, and may even step it up to the $5 level, but I am not comfortable with more than that variance.

Actually, if I ever decided to step up my betting amount, I would likely multi-hand instead of increasing my one-spot bet. For example, 3-spotting $2 has about the same variance of single-spot betting about $3.50. The only issue is whether I can play more (or a similar number) of hands per hour 3-spotting. Maybe I'll try that on my next bonus and see how it goes.

My spreadsheet can also show me how lucky I was to be up $250 on my first bonus. As it turns out, there is a 5% chance the way I played that bonus that I would be up $250 (or more). Always nice to hit the 20-1 shot on my first try. :-)

My next post may be about how the EV of differing betting strategies is not quite the same, but will also so how little EV can be gained by them, at the expense of generating insane amounts of risk.