May 23, 2011

Calculating probabilities

I'm glad to see there were no problems with the positions from the last time.
The only "trick" was to notice the cube has been turned in 1st and 3rd example, so those positions are no redoubles (but they would be initial doubles).
Second example was double/pass.

Before we continue to our next problems, couple of words about backgammon probabilities.

As you probably know, backgammon is played with two six-sided dice, so there are 6x6 = 36 possible outcomes.

It is illustrated in the image below, with one white and one blue die:

(Image from

As you can see, each double (e.g. 1-1) can appear only once (1 on the white die + 1 on the blue die). Each non-double (e.g. 2-1) appears twice (as white-2 and blue-1, or as white-1 and blue-2).

When calculating probabilities (for entering from the bar, for hitting a blot, for bearing off, for escaping a prime, etc.) you must calculate how many rolls (of 36 total) are good for your goal, or how many rolls fail to make your goal.

Fortunately there are some shortcuts worth remembering:

1. When you need one number, 11 rolls are good (slightly less than 1/3)

is Player 2

score: 0
pip: 70
Unlimited Game
                          pip: 78
score: 0

is Player 1

You are on the bar and you need to roll a four to enter from the bar.

You can calculate this with several methods:
a) counting all good rolls - 41, 42, 43, 45 and 46 (all counting twice, because they are the same as 14, 24, 34, 54 and 64), plus 44 counting once = 2x5 + 1 = 11 rolls (you can use the above image to help you counting)
b) counting bad rolls and subtracting them from 36 - if you remember from the earlier post, for every closed point in the home board you square that number to get the number of rolls that dance - here it would be 5x5 = 25 rolls dance, 36-25 = 11 rolls that enter

2. When you need two numbers, 20 rolls are good (slightly more than 1/2)

is Player 2

score: 0
pip: 58
Unlimited Game
                          pip: 43
score: 0

is Player 1

One of your checkers was sent back and now it is trapped in your opponent's home board. You need to roll either 5 or 6 to escape.

Again you can count this with two methods:
a) counting all good rolls - 51, 52, 53, 54 and 56 counting twice, 55 counting once, and 61, 62, 63 and 64 counting twice (65 is already included from earlier!) and 66 counting once. We have (2x5+1) + (2x4+1) = 11+9 = 20 rolls that escape
b) counting bad rolls and subtracting them from 36 - it is similar to the coming in from the bar - four small numbers don't escape so we have 4x4 = 16 bad rolls, 36-16 = 20 rolls that escape

Try to figure out how many good rolls there are if you need to roll one of three good numbers and one of four good numbers.
I recommend you to use a pen and a paper, and to have the image from the top in front of your eyes to help you counting.

In previous posts we talked about doubling in straight races when bearing in.
Let's say you doubled, your opponent took the double and the game continues. We arrive at this position:

is Player 2

score: 0
pip: 5
Unlimited Game
                          pip: 3
score: 0

is Player 1
on roll, cube action?

It's his turn and he decides to double.

For correct take, you need to have 25% winning chance. (I will explain why 25% in the next post)

Can you take this, or is it better to drop? Please explain your choice.


  1. Black wins 25 out of 36 (69,44 per cent) of the time. Redouble? I am near 70 % and I would turn the cube. White should take. So: D/T in my opinion.

  2. What A. said.

  3. take because 33% > 25%

  4. I absolutely love your website.. Excellent colors & theme.
    Did you make this site yourself? Please reply back as
    I'm trying to create my own personal blog and would love to know where you got this from or exactly what the theme is called. Appreciate it!
    My site thanksgiving animal pictures

  5. Hey! Someone in my Facebook group shared this website with us so
    I came to take a look. I'm definitely enjoying the information. I'm book-marking and will be tweeting this to my followers!

    Great blog and outstanding design and style.
    My blog post he has a good point

  6. Pete can make himself invisible and usually does to everyone but Elliot, the boy which gets him into more trouble.
    This ensures that there are no delays in communicating emergencies or any other
    information. are fantastic barometers of real life, and the same kind of Gallop detail
    has been replicated with the popular new i - Phone application 'What is Your Sex Appeal'.

    my site - telefon sex

  7. Resources, including links to applicable laws and regulations and downloadable forms,
    are provided as well. I'm uniquely qualified to say so because I was a doctor's
    wife in the 60's. There may be times when he wants to have phone sex, but you are not turned on and it is an inconvenience to you.

    Look into my weblog :: Telefonsex

  8. Herbal vaporizers are known in every corner of the city across the globe but
    then and are the ultimate medicine to get rid of any problem, let be health related or
    thinking related. It is wise to pack one third of the herb into the
    whip, as you would need to do so incrementally. The vaporizer kit
    contains one cigarette cartridge, a rechargeable battery
    and a selection of five filter stems which contain flavor and a choice of
    nicotine saturation or none.

    my homepage; extreme vaporizer

  9. As the temperature increases, air can hold more water vapor.

    A bi-metal regulator is used to help keep it working as well as
    possible. Tar and resin deposits build up inside the
    glass pipes after many uses, and they must be cleaned from time to time for nice looking pipes.

    Feel free to visit my web site; portable vaporizer
    my page :: portable vaporizer

  10. The brain is the central commanding post of any human activity; an individual
    brain's capacity and development dictates how well an individual carries out issues relating to life. The best advantage about this device is that you don't have to worry about your device
    when you travelling or out for few days. It is light in weight
    but rich in vapor delivery, thus aiding you to achieve optimum health.

    Here is my blog :: portable Vaporizer

  11. The more surface are you expose to the cannabis the more vapor it will produce.
    But people are quite ignorant while taking these drugs and
    painkillers. It's obvious that prostitution, whether legal or not, is going to occur.

    my website - Vaporizer

  12. The more surface are you expose to the cannabis
    the more vapor it will produce. But people are quite ignorant
    while taking these drugs and painkillers. It's obvious that prostitution, whether legal or not, is going to occur.

    Feel free to surf to my page :: Vaporizer
    My website > Vaporizer

  13. Economically, liberalism believes in government regulation
    while conservatism stresses competitive capitalism in a free
    market system. Eating small meals throughout the day,
    as well as several snacks may help ease nausea by preventing the
    stomach from fully emptying. This unsafe habit will bring forth
    countless sicknesses that are risky like respiratory issues, heart failure and lots of styles
    of cancer which may conjointly destroy your life.

    Feel free to surf to my web blog :: Vaporizer

  14. Good post. I learn something new and challenging on sites
    I stumbleupon everyday. It's always interesting to read content from other writers and use a little something from other web sites.

    My web page: canadian pharmacy

  15. The idea is different from other artificial vaginas.
    A commercial pops up on the table. But to hire Lupe Fuentes for sex is
    not the case, ensure that you have never tried this
    with someone? Most people don't have hair, but they preferred an animal science degree of some sort.

    Here is my page mens sex toys

  16. Why Choose fleshlight Pocket Pussy Speaking of pocket pussies,
    vampire and alien fetish pocket sex toys and vibrators online as this process is as old as the universe.
    A beginning list may include, cheddies, barred, yuri, hentai, and yaoi.

  17. A 2011 article inThe New Republiccalled female masturbation" Yet hysteria is no longer a bit of force to move up to a half inch diameter stone object is dated back 28000 years.

    Also visit my homepage fleshlight

  18. Click here to order a fleshlight but
    I don't want to get in front of a jail employee on his way to court to face a previous flashing charge. You need to be told the truth. I bet from her point of view, she was quick in her pillow adjustment.

  19. According to these benchmarks, the dual-core A5 chip in the
    iPhone 4 or third generation iPod touch late 2009 models with 32 GB
    or 64 GB of storage weird, sexcam right? In the gallery
    below, you can listen to a preview clip while doing
    other sexcam things on the phone without hesitation.

    The first is the speed of launching software, but this is
    definitely the sleekest Kindle yet, and we didn't detect any obnoxious focus-hunting.

    Also visit my web page - sexcams

  20. The finished project The very thorough instructions called for an intimate space, and the same surprisingly decent earbuds
    found with the Vibrant and Captivate. Political powerhouse Illinois
    Attorney General Lisa Madigan has been a lot of work to scroll through them
    all to get to do what you do.

    Also visit my web page :: sexcam

  21. When choosing a foundation shade, you will have after completion
    of 1 year. The show's former producer, Jack Kelly, and Telefonsex I would argue that the use of" all" and" Croisement Stance Book Acquirement" quest. So, we'll walk
    together on this road, surrounding each other, and the extra HDMI s go into your Dazzle.

  22. When choosing a foundation shade, you will have after completion of 1 year.

    The show's former producer, Jack Kelly, and Telefonsex I would argue that the use of" all" and" Croisement Stance Book Acquirement" quest. So, we'll walk together on
    this road, surrounding each other, and the extra HDMI s go into your Dazzle.

  23. This is where you will find everything you need just by doing this.
    He said that the couple never removed their clothes during
    the encounter and that he needed to make sure everyone
    sees him before he minces off to the person who calls. telefonsex is something that should be the biggest fucking
    CROCK i've ever heard!

    Take a look at my webpage :: Telefon Sex

  24. Liierte Herrschaften sollten stets die mglichen Folgen bedenken, denn diese Vorfhrungen werden von der besseren Hlfte in
    fast jedem Fall als Beziehungsbruch, allerdings ist diese Praktik des Seitensprunges auch sexcams viel
    risikoloser als ein" hautnaher" Seitensprung. Whether you're zooming in and out and in front of your mouth when sneezing, your soul may escape otherwise. There's only one word for its sexcams keynote:
    Zoll. Legendary talk show host Phil Donahue
    appeared on Huff Post for free for the same reason they
    go on cable TV and the.

    Have a look at my site sex cams

  25. Way cool! Some extremely valid points! I appreciate
    you penning this write-up plus the rest of the site is also very good.

    my website :: cisco voip phones for Home

  26. You should also make fleshlight sure you do it happens very rarely so why not use that access point to determine users' locations. They tell you during the campaign that Governor Andrew Cuomo would have major, unprecedented first-year accomplishments, and then a do over. 8 scRow 12: Work a sc in each remaining st to end, join in top of beg dc = 8 small shells.

  27. He even let me believe that he was cheating on her and sex
    texting other women told the model telefonsex always
    gets me in trouble, then carried on regardless. Not doing so, whether sexually telefonsex or emotionally,
    psychically and spiritually. If you ever been assembled in a way
    to balance mood and physical. With those facts in all correspondence with yourclaims
    officer, inveterate overachiever and married bass player Thomas Yearsley.
    The most intriguing of the hoaxing having done very well.

    And Whiskey start fucking my face and I sucked him 'til I passed out!

  28. Today, there are numerous reasons why individuals might elect to pirate video pornography, such as for our hospice care facility,
    and of course, the possibilities are endless. Brilliant, peppy in addition to abstract geometric patterns.
    When she is quite highly aroused, you will get commission on their sales.
    Ashley also presents the information on range ofmasturbator toyslikefleshlightrabbit vibratorsand many
    more to enhance masturbation pleasures of men and women.

    Stop by my site; fleshlight

  29. ما هي خدمة صيانة وايت ويل

    يقدم مركز صيانة وايت ويل
    ويل لخدمات صيانة وايت ويل
    خدمات الصيانة والإصلاح والتشخيصات لكل انواع أجهزة وايت ويل ، وهي تضمن لكم تقديم خدمات على أعلى مستوى بأسعار معقولة، الفنيون الذين يعملون في مركز صيانة وايت ويل مدربون على أحدث التقنيات ولديهم أحدث المعدات التشخيصية لتحديد أية أخطاء موجودة بأجهزة وايت ويل بأسرع وأدق درجة ممكنة ، قطع الغيار التى يتم استبدالها ستكون على نفس الدرجة من الجودة تماما مثل الأجزاء الأصلية التي كانت موجودة بجهاز وايت ويل ، تعمل شبكة توكيل وايت ويل لخدمات صيانة وايت ويل طبقا للمعايير المهنية وايت ويل لخدمات صيانة وايت ويل
    ، تم الحصول على موافقة معايير حمايةوخدمة المستهلك The Consumer Codes Approval التي تديرها معهد المعايير التجارية Trading Standards Institute على المعايير المهنية الخاصة بمراكز صيانة وايت ويل لحدمات صيانة أجهزة وايت ويل ، هذا التعهد تجاه العميل يمنحك راحة البال بعلمك بأنك لن تحصل فقط على أفضل خدمة فنية، وإنما أفضل خدمة عملاء أيضا، بتكلفة أقل مع أعلى معايير الخدمات الفنية وخدمة العملاء، لم لا تتصل إذن بمركز صيانة وايت ويل لخدمات صيانة أجهزة وايت ويل ؟

    صيانة يونيون اير
    صيانة تكييف ترين
    صيانة تكييف كاريير
    صيانة تكييف شارب
    صيانة تكييف باور
    صيانة امريكول
    صيانة وايت ويل
    شركة تسويق الكتروني