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Do I set up 55? What could be more simple?

Posted By: Nack Ballard
Date: Saturday, 6 February 2010, at 4:02 a.m.

In Response To: Do I set up 55? What could be more simple? (Steve Mellen)

By my calculations, Black gets off in 2 rolls only 508 times out of 1296. Since you are a fairly big favorite to get 2 more rolls, OTB I would make the play that caters to that scenario rather than playing for one roll, although I suspect it's pretty close.

But then I sat down with pen and paper and calculated that the 6542 formation gets off in two rolls 238 times out of 1296, and the 5543 formation gets off in two rolls 223 times out of 1296. That's not much of a difference! Seems clear to me that a play which gives up 36 wins (out of 1296) on the first shake in order to gain 15 wins (out of 1296) on the second shake, assuming you even get a second shake, must be wrong. So play for 55.

I appreciate you doing the calculations. In general, it makes it easier for the second person (me) to come around and do the same. My counts differ as follows:

Out of 1296, I have the 6542 formation bearing off 350 times, 5543 bearing off 322 times, and Opp's 6222 bearing off either 497 or 499 times (depending on how he plays the 31 roll).

To help you find your error(s) and/or mine, I'll list the number of offs (in parentheses) after each roll. Remember to double the non-doublets.

6542 formation: 65 (23), 64 (19), 63 (5), 62 (10), 61 (4), 54 (13), 53 (4), 52 (8), 51 (4), 43 (4), 42 (6), 41 (3), 32 (3), 31 (3), 21 (3); 66 (36, 55 (36), 44 (36), 33 (5), 22 (10), 11 (3). Total = 350.

5543 formation: 65 (17), 64 (14), 63 (10), 62(4), 61 (4), 54 (14), 53 (10), 52 (4), 51 (4), 43 (4), 42 (4), 41 (3), 32 (3), 31 (3), 21 (3); 66 (36), 55 (36), 44 (36), 33 (5), 22 (4), 11 (3). Total = 322.

6222 formation: 65 (25), 64 (25), 63 (25), 62 (25), 61 (5), 54 (5), 53 (5), 52 (5), 51 (5), 43 (5), 42 (5), 41 (5), 32 (5), 31 (4 or 5), 21 (4); 66 (36), 55 (36), 44 (36), 33 (36), 22 (34), 11 (13). Total = 497 or 499. Let's call it 498.

So, with 6/5, Yellow bears off 1 extra time in 36 on one roll. The 1/36 comes into play the 498/1296 times Brown would have borne off in two rolls, minus the 1/36 he bears off in one roll. The gain is therefore (36*498 - 36) / 1296^2, which is 17892 / 1296^2.

In exchange (with 6/5), Yellow gives up 350 - 322 = 28/1296 in two rolls. This comes into play the 798/1296 that Brown is unable to bear off in two rolls (at this point, you can already see it's going to be bigger). The loss from 6/5 is therefore 28*798 / 1296^2, which is 22344 / 1296^2.

The net loss from 6/5 is therefore 4452 / 1296^2, which is 3.44 / 1296 or 0.27%. I would imagine that three-roll non-recube situations are negligible, though if anything they further (comparatively) diminish the worsely distributed 6/5.

The only factor remaining is Yellow's recube vig when Brown fails to get off in two rolls. Brown's bad sequences are 41 32 31 with 61 51 41 31 21, 21 with 41 32 31 21 11, and 11 with 31 11 21, which means he fails to get off in three rolls with 2*10*10, 2*10*2, 2*8*10, 2*2*1, 2*2*10, 2*2*2, 2*2*11, 2*2*9 and 2*1*1, totalling 534 / 36^3 = 6.9%. That is parlayed with...

By playing 6/5, Yellow is in a position to gain (from cubing) by rolling 54 in most of the scenarios above and getting off with 14 instead of 13 numbers; i.e., 2*1/1296. However, she gains by rolling 65 and getting off with 23 instead of 17 numbers, or by rolling 64 and getting off with 19 instead of 14 numbers, in all scenarios above; i.e., 2*6/1296 + 2*5/1296 = 22/1296.

Therefore, the recube-gain scenarios also favor 3/2, by roughly 20/1296 * 6.9% = 0.1%. Fortunately, it is unnecessary to orient ourselves to exactly what that is worth because 3/2 is already ahead of 6/5 (by 0.27%).

In summary, 6/5 gains only on the 1-roll scenario of double 5s. The gain from the two-roll scenarios from 3/2 overshadow that (relatively speaking), and anything gained from three-roll scenarios (recube or not) are gravy.

If your 238, 223 and 508 numbers are correct, the parallel (though abbreviated) calculation runs something like this:

With 6/5, Yellow bears off an extra 1/36 in one roll, parlayed with the 508/1296 Brown bears off in two minus the 1/36 he bears off in one, for a gain of (36*508 - 36) / 1296^2, which is 18252 / 1296^2.

In exchange, Yellow gives up 238 - 223 = 15/1296 in two rolls, parlayed with the 788/1296 Brown can't bear off in two, for a loss of 15*788 / 1296^2, which is 11820 / 1296^2.

Netting the two paragraphs yields 6432 / 1296^2 = just under 5/1296, or 0.38% in favor of 6/5. It is still necessary to consider third roll scenarios (recube and no recube). These surely won't erase the 0.38%, though.

In the end, it comes down to whether your 238 and 223, or my 350 and 322, are closer to the true bear-off-in-two counts (as the difference between your 508 and my 498 has relatively little impact). Your counts led you to conclude 6/5 is better; my counts led me to conclude that 3/2 is better.

As Neil said, "What could be more simple?"

Nack

Messages In This Thread

• Do I set up 55? What could be more simple?
  neilkaz -- Friday, 5 February 2010, at 8:31 p.m.
  • Do I set up 55? What could be more simple?
    Stick -- Friday, 5 February 2010, at 8:37 p.m.
    • Do I set up 55? What could be more simple?
      Matt Cohn-Geier -- Friday, 5 February 2010, at 10:14 p.m.
  • Do I set up 55? What could be more simple?
    Bob Koca -- Friday, 5 February 2010, at 9:33 p.m.
    • Do I set up 55? What could be more simple?
      Steve Mellen -- Friday, 5 February 2010, at 11:00 p.m.
  • Do I set up 55? What could be more simple?
    Timothy Chow -- Friday, 5 February 2010, at 10:03 p.m.
  • Do I set up 55? What could be more simple?
    Steve Mellen -- Friday, 5 February 2010, at 10:49 p.m.
    • Do I set up 55? What could be more simple?
      Nack Ballard -- Saturday, 6 February 2010, at 4:02 a.m.
      • Do I set up 55? What could be more simple?
        Steve Mellen -- Saturday, 6 February 2010, at 5:28 a.m.
        • Do I set up 55? What could be more simple?
          Nack Ballard -- Saturday, 6 February 2010, at 5:39 a.m.
          • Postscript
            Steve Mellen -- Monday, 8 February 2010, at 6:20 p.m.
            • Postscript
              Bob Koca -- Monday, 8 February 2010, at 8:49 p.m.
            • Postscript
              Nack Ballard -- Tuesday, 9 February 2010, at 7:24 a.m.
      • Do I set up 55? What could be more simple?
        Matt Cohn-Geier -- Saturday, 6 February 2010, at 9:36 a.m.
  • OTB etc
    neilkaz -- Monday, 8 February 2010, at 8:56 p.m.