Daniel Murphy did such a great job covering this thread I don't really need to do it if you read his post. I'm going to go over it anyway because I may add a comment or two that I don't realize right now. First, the problem again:
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| XGID=--B---------------------b-:1:1:1:00:2:0:0:5:10 | ||||
 on roll, cube action? | ||||
is Player 2
The first thing we should figure out is our opponent's take point.
- Double/Pass would give us -5 -1 Crawford or 16% match equity
- Double/Take/Lose would lose the match or 0% match equity
- Double/Take/Win would give us -1 -3 Crawford or 75% match equity
Using these numbers into the formula for take points, risk/(risk+gain) would convert to 16/(16+59) or 16/75 which yields 21.3% as the raw or dead cube take point.
Here since it is a last roll situation that's all we need, the raw take point. No adjustments for gammons or recube vig because we roll and game over. Now that I have my opponent's raw take point I would move on to estimating how often I win this game. In this problem it's simple, I win on any roll that doesn't contain an ace but I do win on 11. That's 26/36 rolls where I win which converts to 72.2%. You could also look at it the other way, how often does your opponent win. He will win on the 10/36 times you miss or 27.8%.
Our opponent's take point is 21.3% and he has 27.8% so he ha a clear take. Now, do we have a cube? Are we in the doubling window? At the time before having this question put to me I didn't remember this formula. I had seen it before and realized how useless in general it was. It turns out that between the 3 players I polled, myself, John O'Hagan, and David Rockwell and all our years of combined experience we have actually had the need to use this OtB twice. If you don't commit it to memory you won't be missing out on much but as I'm to the point where learning new useless things should be my specialty I thought I'd commit it to memory esp. since it is that easy and I already knew it.
Again for when the doubling window opens we can use risk/(risk+gain). In this example the risk is 25% and the gain is 16% so 25/(25+16) or 25/41 which is 61%. We clearly have more than 61% on this very volatile last roll situation so we double.
There was also discussion in this thread on what is easier to memorize, a match equity table or take points. It is more useful to know the match equity table at least out to 5 away 5 away and then after that you can use Neil's numbers for basically everything else.
When I started learning everything under the sun that is backgammon it was a very daunting task. There was so much to learn, I didn't know where to start, and it seemed like numbers were flying everywhere, formulas that I wasn't familiar with, math is not my forte so it was an even bigger put off. I finally broke down by breaking it down into small segments and learning, actually learning, a little at a time. Committing it to memory, adding on and reviewing it the next time. For the match equity table I first tried to understand it by looking at an overview of it.
As was mentioned, a 5 away 5 away match equity table is only 25 cells 5 of which you know already. Of the 20 left you only need to know 10 of them as its counterpart score reflects its equity. Take a look.
| 1 | 2 | 3 | 4 | 5 | |
| 1 | .50 | ||||
| 2 | .50 | ||||
| 3 | .50 | ||||
| 4 | .50 | ||||
| 5 | .50 |
Scaled down it doesn't look so scary does it? How I personally started to fill in the blanks wasn't straight memorization. I don't claim this to be the best method but it's what I did in the beginning. I next thought to myself 'how is the equity for -1 -2 Crawford figured?' Well, I will have to win two games in a row and I'm 50% in either game so 0.5 x 0.5 would give me 0.25 or 25% chances from -1 -2 Crawford. However I can also win a gammon in the first game and win the match so my equity has to be higher than that 25%. Turns out that adds ~7% on to my match winning chances. That makes sense, so now I can both understand and fill those two blocks in on my mental match equity table.
| 1 | 2 | 3 | 4 | 5 | |
| 1 | .50 | .68 | |||
| 2 | .32 | .50 | |||
| 3 | .50 | ||||
| 4 | .50 | ||||
| 5 | .50 |
Next on my list, 1 away 3 away Crawford. This is really simple. Now you do need to win two games in a row to win the match because gammons in your first game really aren't worth anything. You can also ignore the minute backgammon wins in your first game when figuring out the match equity. So you need to win two games in a row, 0.5 x 0.5 or 0.25, 25% as stated above. Forging ahead:
| 1 | 2 | 3 | 4 | 5 | |
| 1 | .50 | .68 | .75 | ||
| 2 | .32 | .50 | |||
| 3 | .25 | .50 | |||
| 4 | .50 | ||||
| 5 | .50 |
Branching out we now tackle 2 away 3 away and 3 away 2 away. This clicked immediately in my head because the match equities are 60%/40% and the ratio 3:2 is 60/40. Next!
| 1 | 2 | 3 | 4 | 5 | |
| 1 | .50 | .68 | .75 | ||
| 2 | .32 | .50 | .60 | ||
| 3 | .25 | .40 | .50 | ||
| 4 | .50 | ||||
| 5 | .50 |
Now it should feel like you've made progress. I will spare you the details of the rest I switched to memorization for the most part.
A couple points mentioned in this thread and always worth repeating:
- In a match between games you should mentally note the score and what your goals are at this score. It could be as simple as 'it's -11 -11, I'm playing like money until I see a 4 cube pop out of the hat' or -3 -3 where if you don't already know the take points and gammon values you could start figuring them out in your head on your down time.
- The more you do something the easier it becomes. Mental arithmetic can seem tough to straighten out for those of us who aren't accustomed to it. Juggling all these numbers around trying to keep straight what we just figured out two seconds ago isn't easy at first. Force yourself to do it and do it again and again and again and before you know it, it'll be second nature. Much like keeping a running pip count from the beginning of the game where most people say 'that's too distracting, I can't do it fast enough, etc...' it's actually quite easy if you take a little bit of time and put forth the effort. In a world hopefully full of clocks soon, these things give you quite a boost.
Stick
***A special thanks to Tom Keith/backgammon galore for providing such beautiful tables for me to thief when I need a piece of code.