The Kleinman Probability

The Kleinman probability (expressed as a percentage between 0 and 100%) allows you to assess the relative strength of your position in a running game.

To display the Kleinman probability in GridGammon, just add the Kleinman toolbar item to your Board window.

  1. Open the Board window (menu Window > Show > Board)

  2. Open the window's toolbar setup sheet (View > Customize Toolbar...)

  3. Drag the Kleinman item to your toolbar

  4. Click Done.

NOTE: GridGammon will only display the Kleinman probability under the following conditions:

  1. The players are in a no contact position. The probability is only meaningful in a running game.

  2. The person rolling is leading. If you are rolling, your pip count must be less than [your opponent's pip count + 4] (see below for more details).

  3. The sum of the two player's pip counts is greater than 100. When below this threshold, other game factors reduce the accuracy of the computation.

These conditions mean that the Kleinman probability is only significant (and hence displayed) for a small portion of a game. Many games will have few if any turns where Kleinman is shown. This is not a software bug, but an indication of the narrow conditions required for this analysis to be useful.

Computing the Kleinman Probability

If you're interested, here are the technical details about how the Kleinman probability is computed. First we calculate a related number known as the Kleinman ratio. This ratio K, is calculated by the formula:

K = (D * D) / S

where D = the pip count difference of the two players, plus 4 to compensate for the player on roll's additional pips, and S = the pip count sum, minus 4 to compensate for the player on roll.

The probability, P, calculated from K as follows:

If K > 1.0, then P = 0.76 + 0.114 * ln(K)
If K < 1.0, then P = 0.5 + 0.267 * sqrt(K)

Restricted to the range [0.0, 1.0], P is then displayed as a percentage between 0 and 100%.

The Kleinman ratio was defined by Danny Kleinman, author of numerous books on backgammon emphasizing mathematical analysis. For more information, you can contact him at 5312 1/2 Village Green, Los Angeles, CA 90016.

Thanks to Øystein Johansen for the formula to convert the Kleinman ratio to a probability, which he described in a posting to the rec.games.backgammon newsgroup in February 2001.