JCT: In my last post, I reproduced the complete array of
threat outs vectors: (best aligned in txt mode).
Odds for Outs Danger vs Opponents
Opponents
1 2 3 4 5 6 7 8
Outs
2 22 11 7 5 4 3 3 2
4 10 5 3 2 2 1 1 1
6 7 3 2 1 1 1 2 2
8 5 2 1 1 2 2 3 4
10 4 2 1 2 2 3 5 6
12 3 1 2 3 4 5 7 10

14 2 1 2 3 5 8 12 18
16 2 1 3 5 8 12 19
18 2 2 3 6 11 19
20 1 2 5 9 16
24 1 4 8 18
30 2 7

Sorry but I've had to make some changes:

Opponents
1 2 3 4 5 6 7 8
Outs
2 -22-10-7 -5 -4 -3 -2 -2
4 -10-5 -3 -2 1 1 1 2
6 -7 -3 -2 1 1 2 2 3
8 -5 -2 1 2 2 3 3 4
10 -4 1 1 2 3 4 5 7
12 -3 1 2 3 4 6 8 11

14 -2 1 2 4 5 8 12 18
16 -2 2 3 5 8 12 19 >22
18 -2 2 4 7 11 19 >22
20 1 3 5 9 17 >22
23 1 3 7 15 >22
30 2 8 >22

As the number of opponents goes up, each vector going left
to right is basically odds in your favor, even money, odds
against you. I have to distinguish between for and against
with a negative sign.

Since the important odds I'll need to remember are when I'm
chasing, I've let them be positive. As to the odds when I'm
ahead that I don't need to remember, when I'm ahead by 10%
or 90%, I've let them be negative, the odds of being beaten.

So the negatives are odds against you being beaten, the
positives are odds against you being good.

But unfortunately, I rounded off the original odds. That was
a mistake. When I'm chasing, they should be rounded up, when
I'm being chased, they can be rounded down. It's not too
important when I'm in the lead but it is when I'm chasing.

For instance, looking at the old 10 outs vector

Against 5 opponents, the actual value is 2.4:1 rounded down
to 2:1. Let's say I accept 20:10 on my 24:10 shot. I'll lose
one 24 times and win two 10 times: (20-24)/34 = -11%! So I
really needed to round up to 3 to stay in the black.

Against 6 opponents, it's actually 3.4:1 rounded down to 3.
Let's say I accept 30:10 on my 34:10 shot against 6
opponents. I'll lose one 34 times and win three 10 times:
(30-34)/34 =-9%! So I really needed to round up to 4.

Opp= 1 2 3 4 5 6 7 8
10 -4 -2 1 2 3 4 5 6

Nice, eh? 1 2 3 4 5 6

Similarly, if the odds are with me, I should have rounded
down though it isn't very important.

Take 2 outs against 2 opponents which was rounded up from
the real 10.8:1. to 11:1. Seems logical to round up to 11.
That's 108:10. But if I win 1 bet 108 times and lose 11 bets
10 times, (108-110)/118 =-1.6%. -1.6% may not seem much but
it's more than the vig at Craps.

So I rounded up the numbers when I was behind. And I rounded
down most of the time when I was in the lead except when I
was within a few percent for form.

So the board threat tool has been upgraded from dealing with
only 1 opponent to now deal with up to 8 just in time for me
to include it as the bonus in my Turmel-Two-Step book with
15,000 exercises that I'm finally going to publish.

Notice that the 1 opponent vector is Poker Power Tool #1!
Outs Odds
2 -22
4 -10
6 -7
8 -5
10 -4
12 -3
14 -2

Notice the symmetry in small percentage threats for vectors
for 2 outs across or 1 oppt down:

Opp= 1 2 3 4 5 6 7 8
2 -22-10-7 -5 -4 -3 -2 -2 and

Opp= 1
Outs Odds
2 -22
4 -10
6 -7
8 -5
10 -4
12 -3
14 -2
16 -2

Also, vectors for 4 outs across or 2 opponents down

Opp= 1 2 3 4 5 6 7 8
4 -10-5 -3 -2 1 1 1 2

Opp= 2
Outs Odds
2 -10
4 -5
6 -3
8 -2
10 1
12 1
14 1
16 2

And for vectors for 6 outs across or 3 opponents down:
Opp= 1 2 3 4 5 6 7 8
6 -7 -3 -2 1 1 2 2 3

Opp= 3
Outs Odds
2 -7
4 -3
6 -2
8 1
10 1
12 2
14 2
16 3

Even vectors for 8 outs across or 4 opponents down:
Opp= 1 2 3 4 5 6 7 8
8 -5 -2 1 2 2 3 3 4

Opp= 4
Outs Odds
2 -5
4 -2
6 1
8 2
10 2
12 3
14 4*
16 5*

Checking 10 outs across or 5 opponents down:

Opp= 1 2 3 4 5 6 7 8
10 -4 1 1 2 3 4 5 7

Outs
2 -4
4 1
6 1
8 2
10 3
12 4
14 5
16 8*

And we would expect that 4 outs against 2 opponents would be
the same as 2 outs against 4 opponents or 8 outs. It's only
when you get into the big numbers that things don't add up
exactly.

I've also decided to use 23 outs instead of 24 for the
purposes of explaining it. 23 outs is half of 46 cards so
it's easier to explain how you'll lose 23/46 or 1/2 the time
with 1 opponent, it's squared with 2 opponents losing
(1/2)^2 = 1/4, 3:1; it's cubed with 3 opponents, 1/8, 7:1;
with 4 opponents, it's 1/16, 15:1. with 5 opponents it's
1/32, 31:1; with 6 opponents it's 1/64, 63:1; etc. So it's
easier to explain how the chances of winning shrink as 1/2;
1/4; 1/8; 1/16, 1/32 using 23 outs instead of 24.

And of course, after handling 1/2 of the deck hurting you, I
use 30 outs or about 2/3 of the deck hurting you and 1/3
helping. So you stay good against 1 opponent 1/3 (2:1) of
the time and against 2 opponents, it's squared (1/3)^2 = 1/9
(8:1). With 3 opponents, it's cubed (1/3)^3 = 1/27 (26:1).

So the odds you really need to remember are:
Opponents
1 2 3 4 5 6 7 8
Outs
2 + + + + + + + +
4 + + + + + + + 2
6 + + + + + 2 2 3
8 + + + 2 2 3 3 4
10 + + + 2 3 4 5 7
12 + + 2 3 4 6 8 11

14 + + 2 4 5 8 12 18
16 + 2 3 5 8 12 19
18 + 2 4 7 11 19
20 + 3 5 9 17
23 + 3 7 15
30 2 8

So, this should be the final revision of these vectors.

Opponents
1 2 3 4 5 6 7 8
Outs
2 -22-10-7 -5 -4 -3 -2 -2
4 -10-5 -3 -2 1 1 1 2
6 -7 -3 -2 1 1 2 2 3
8 -5 -2 1 2 2 3 3 4
10 -4 1 1 2 3 4 5 7
12 -3 1 2 3 4 6 8 11

14 -2 1 2 4 5 8 12 18
16 -2 2 3 5 8 12 19 >22
18 -2 2 4 7 11 19 >22
20 1 3 5 9 17 >22
23 1 3 7 15 >22
30 2 8 >22

You have to admit, this sure adds an elegance to the board
threat tool now that we can handle not only the threat from
1 opponent but also from 8.





--
Abolitionist Debt Slave Leader John C."The Banking Systems Engineer"
Turmel for UNILETS interest-free time-based currency in U.N. resolution
C6 to Governments in the http://www.un.org/millennium/declaration.htm
http://www.cyberclass.net/turmel USENET blog: alt.fan.john-turmel