11-02-2005, 06:59 AM
{Apologies in advance if this happens to have already been covered on a past topic in the archives.}
I’ve been revisiting the “A Simple Technique” chapter from Sklansky’s “Fighting Fuzzy Thinking” book from a few years ago. In this chapter, he provides a cool technique for computing actual odds in Hold ‘Em with two cards yet to come.
It got me to thinking about how to simplify “A Simple Technique.” I thought back to blackjack card counting systems, in which the numbers assigned to various cards (e.g, +1, 0, -1) are sometimes actually quite different from the actual card values computed by the late great Peter Griffin in “The Theory of Blackjack.”
As it turns out, the card counting systems work great because the card values assigned are reasonably accurate, i.e., they introduce only minor acceptable errors, and are much easier to use in the heat of battle. So I took this mindset and tried to apply it to “A Simple Technique” and came up with something pretty cool.
Starting with the 1980 total hands and rounding it up to 2000 introduces only minor errors. The computation is the same as the rational function y=1/x. In the spreadsheet I plotted several points of computed odds, starting with both 1980 and 2000. As you can see the error is almost non-existent.
Here are the brute force numbers from my excel worksheet:
Odds Odds
Outs 1980 2000
25 78.2 79.0
50 38.6 39.0
75 25.4 25.7
100 18.8 19.0
125 14.8 15.0
150 12.2 12.3
175 10.3 10.4
200 8.9 9.0
225 7.8 7.9
250 6.9 7.0
275 6.2 6.3
300 5.6 5.7
325 5.1 5.2
350 4.7 4.7
375 4.3 4.3
400 4.0 4.0
425 3.7 3.7
450 3.4 3.4
475 3.2 3.2
500 3.0 3.0
525 2.8 2.8
550 2.6 2.6
575 2.4 2.5
600 2.3 2.3
625 2.2 2.2
650 2.0 2.1
675 1.9 2.0
700 1.8 1.9
725 1.7 1.8
750 1.6 1.7
775 1.6 1.6
800 1.5 1.5
825 1.4 1.4
850 1.3 1.4
875 1.3 1.3
900 1.2 1.2
925 1.1 1.2
950 1.1 1.1
975 1.0 1.1
1000 1.0 1.0
Now here is the cool part. Borrowing from the "Guesstimation" and other chapters in the book "Mathemagics" (by Arthur Benjamin and Michael Shermer) the goal is to try and simplify the odds computation, once the number of outs is known. The trick is to turn a 4 digit computation into a 3 digit computation without losing any significant accuracy.
Step 1.
Start with the rounded up total number of hands: 2000
Step 2.
Drop the least significant digit, making the 2000 into 200.
Step 3.
Take the sum of total number of outs and round it up or down.
For example, in the first Sklansky example, with Aces and backdoor flush and full house draws against a set of 7's, the total number of outs for the Aces is 248. Round this number up to 250.
Step 4.
Drop the least significant digit, in this example making the 250 into 25.
Step 5.
Make the odds computation: a/b fraction turns into b-a:a odds.
In this example, the 25/200 fraction turns into 200-25:25 or 175:25 or simply 7:1, right on the actual value computed in the book.
Doing a couple of other random examples from the extreme limits of the 1/x function gives:
Example 1, 38 outs:
38 rounded up to 40, drop the zero to give 4.
200-4:4 gives 196:4 or 98:2 or 49:1
This is very close to the actual 51:1 odds produced using 1980-38:38. In the heat of battle, this is a margin of error that I can live with. For all intents and purposes, this would have to be an extremely rare hand (e.g., long odds and a huge pot, or huge implied odds) for me to even consider playing it anyway.
Example 2, 933 outs:
933 rounded down to 930, drop the zero to give 93.
200-93:93 gives 107:93, which is appx 36:31 or appx 7:6 or about 1.17:1
This is also very close to the actual 1.12:1 produced using the actual 1980-933:933. Again, this is a margin of error that I can live with, because in reality, this hand is going to be so close to call that the pot odds are likely going to be good enough for me to play anyway.
While I haven’t run through the whole range of possible computations, I suspect that if the error margins at the extremes are some that I can live with, then the errors between them should be as well.
Further, while I haven’t done the actual computations, I suspect that purposely introducing other minor errors to make numbers easier to divide in the head, will allow one to produce reasonably accurate results even faster.
For example, using the above case of 107:93, rounding 107 up to 108 makes both sides of the odds divisible by 3, making them easily reduce to the 36:31 shown. Then rounding the 31 down to 30 and the 36 to 35, makes both numbers divisible by 5, reducing the odds to 7:6 or 1.17:1.
The 31 could also be rounded up to 32 making both numbers divisible by 4, reducing the odds to 9:8 or again 1.12:1, which is the actual value produced in the full computation.
In conclusion, the key here as in blackjack is not to necessarily get the exact odds value to a couple of decimal places. Rather, the objective is to be able to compute a reasonably accurate odds value, quickly, accurately and consistently, while in the heat of battle. IMHO, these little shortcuts work pretty well and are highly accurate in the typical range of odds computations between the high end of about 20:1 and the low end of about 1.1:1.
I’ve been revisiting the “A Simple Technique” chapter from Sklansky’s “Fighting Fuzzy Thinking” book from a few years ago. In this chapter, he provides a cool technique for computing actual odds in Hold ‘Em with two cards yet to come.
It got me to thinking about how to simplify “A Simple Technique.” I thought back to blackjack card counting systems, in which the numbers assigned to various cards (e.g, +1, 0, -1) are sometimes actually quite different from the actual card values computed by the late great Peter Griffin in “The Theory of Blackjack.”
As it turns out, the card counting systems work great because the card values assigned are reasonably accurate, i.e., they introduce only minor acceptable errors, and are much easier to use in the heat of battle. So I took this mindset and tried to apply it to “A Simple Technique” and came up with something pretty cool.
Starting with the 1980 total hands and rounding it up to 2000 introduces only minor errors. The computation is the same as the rational function y=1/x. In the spreadsheet I plotted several points of computed odds, starting with both 1980 and 2000. As you can see the error is almost non-existent.
Here are the brute force numbers from my excel worksheet:
Odds Odds
Outs 1980 2000
25 78.2 79.0
50 38.6 39.0
75 25.4 25.7
100 18.8 19.0
125 14.8 15.0
150 12.2 12.3
175 10.3 10.4
200 8.9 9.0
225 7.8 7.9
250 6.9 7.0
275 6.2 6.3
300 5.6 5.7
325 5.1 5.2
350 4.7 4.7
375 4.3 4.3
400 4.0 4.0
425 3.7 3.7
450 3.4 3.4
475 3.2 3.2
500 3.0 3.0
525 2.8 2.8
550 2.6 2.6
575 2.4 2.5
600 2.3 2.3
625 2.2 2.2
650 2.0 2.1
675 1.9 2.0
700 1.8 1.9
725 1.7 1.8
750 1.6 1.7
775 1.6 1.6
800 1.5 1.5
825 1.4 1.4
850 1.3 1.4
875 1.3 1.3
900 1.2 1.2
925 1.1 1.2
950 1.1 1.1
975 1.0 1.1
1000 1.0 1.0
Now here is the cool part. Borrowing from the "Guesstimation" and other chapters in the book "Mathemagics" (by Arthur Benjamin and Michael Shermer) the goal is to try and simplify the odds computation, once the number of outs is known. The trick is to turn a 4 digit computation into a 3 digit computation without losing any significant accuracy.
Step 1.
Start with the rounded up total number of hands: 2000
Step 2.
Drop the least significant digit, making the 2000 into 200.
Step 3.
Take the sum of total number of outs and round it up or down.
For example, in the first Sklansky example, with Aces and backdoor flush and full house draws against a set of 7's, the total number of outs for the Aces is 248. Round this number up to 250.
Step 4.
Drop the least significant digit, in this example making the 250 into 25.
Step 5.
Make the odds computation: a/b fraction turns into b-a:a odds.
In this example, the 25/200 fraction turns into 200-25:25 or 175:25 or simply 7:1, right on the actual value computed in the book.
Doing a couple of other random examples from the extreme limits of the 1/x function gives:
Example 1, 38 outs:
38 rounded up to 40, drop the zero to give 4.
200-4:4 gives 196:4 or 98:2 or 49:1
This is very close to the actual 51:1 odds produced using 1980-38:38. In the heat of battle, this is a margin of error that I can live with. For all intents and purposes, this would have to be an extremely rare hand (e.g., long odds and a huge pot, or huge implied odds) for me to even consider playing it anyway.
Example 2, 933 outs:
933 rounded down to 930, drop the zero to give 93.
200-93:93 gives 107:93, which is appx 36:31 or appx 7:6 or about 1.17:1
This is also very close to the actual 1.12:1 produced using the actual 1980-933:933. Again, this is a margin of error that I can live with, because in reality, this hand is going to be so close to call that the pot odds are likely going to be good enough for me to play anyway.
While I haven’t run through the whole range of possible computations, I suspect that if the error margins at the extremes are some that I can live with, then the errors between them should be as well.
Further, while I haven’t done the actual computations, I suspect that purposely introducing other minor errors to make numbers easier to divide in the head, will allow one to produce reasonably accurate results even faster.
For example, using the above case of 107:93, rounding 107 up to 108 makes both sides of the odds divisible by 3, making them easily reduce to the 36:31 shown. Then rounding the 31 down to 30 and the 36 to 35, makes both numbers divisible by 5, reducing the odds to 7:6 or 1.17:1.
The 31 could also be rounded up to 32 making both numbers divisible by 4, reducing the odds to 9:8 or again 1.12:1, which is the actual value produced in the full computation.
In conclusion, the key here as in blackjack is not to necessarily get the exact odds value to a couple of decimal places. Rather, the objective is to be able to compute a reasonably accurate odds value, quickly, accurately and consistently, while in the heat of battle. IMHO, these little shortcuts work pretty well and are highly accurate in the typical range of odds computations between the high end of about 20:1 and the low end of about 1.1:1.