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Bridge Probabilities and Combinatorics



How to Calculate Bridge Bidding Probabilities and Combinations

   To simplify the description below, we will first define a "suit bid". A "suit bid" consists of a number (1, 2, 3, 4, 5, 6, or 7) followed by a suit (Clubs, Diamonds, Hearts, Spades, No Trump). We note that any of the 7 numbers may be combined with any of the 5 suits yielding a choice of 35 possible bids. The bidding process may use none, one, etc., up to 35 of these suit bids. The words "Pass", "double", and "redouble" may also be used as part of the bidding process.

The bidding process is divided into 3 groups:
1) Combinations before the first suit bid.
2) Combinations using suit bids.
3) Combinations after suit bids have concluded.

   For each quantity of suit bids (1-35), the totals for all 3 groups are multiplied together. The grand total is the sum of these plus 1 (For pass, pass, pass, pass).

There are only 4 combinations for group "1)" which are:
Bid
Pass, Bid
Pass, Pass, Bid
Pass, Pass, Pass, Bid

Group "3)" is nearly as simple with only 7 combinations.
Bid, Pass, Pass, Pass
Bid, Double, Pass, Pass, Pass
Bid, Pass, Pass, Double, Pass, Pass, Pass
Bid, Double, Redouble, Pass, Pass, Pass
Bid, Double, Pass, Pass, Redouble, Pass, Pass, Pass
Bid, Pass, Pass, Double, Redouble, Pass, Pass, Pass
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Pass, Pass

   Group "2)" is significantly more complicated. There are 35 possible suit bids (Digits 1 - 7 times 5 suits). Thus, if we only use one of these "suit bids", there are COMBIN( 35, 1) = 35 possible combinations. If we use any 2 "suit bids", then there are COMBIN( 35, 2) = 595 combinations. (Take any 2 from 35). 3 "suit bids" yields COMBIN( 35, 3) = 6,545. This process repeats up through COMBIN( 35, 35) = 1.

In-between each of the suit bids, there are 21 possible intervening sequences:
Bid, Bid      (No intervening "Passes", "doubles", "redoubles")
Bid, Pass, Bid
Bid, Pass, Pass, Bid
Bid, Double, Bid
Bid, Double, Pass, Bid
Bid, Double, Pass, Pass, Bid
Bid, Pass, Pass, Double, Bid
Bid, Pass, Pass, Double, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Bid
Bid, Double, Redouble, Bid
Bid, Double, Redouble, Pass, Bid
Bid, Double, Redouble, Pass, Pass, Bid
Bid, Double, Pass, Pass, Redouble, Bid
Bid, Double, Pass, Pass, Redouble, Pass, Bid
Bid, Double, Pass, Pass, Redouble, Pass, Pass, Bid
Bid, Pass, Pass, Double, Redouble, Bid
Bid, Pass, Pass, Double, Redouble, Pass, Bid
Bid, Pass, Pass, Double, Redouble, Pass, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Bid
Bid, Pass, Pass, Double, Pass, Pass, Redouble, Pass, Pass, Bid

 If there is only 1 "suit bid", the above intervening sequence is used 0 times.
 If there are 2 "suit bids", the above intervening sequence is used 1 time.
 If there are 3 "suit bids", the above intervening sequence is used 2 times.
    etc.
 If there are 35 "suit bids", the above intervening sequence is used 34 times.

Thus the mathematical calculation for the number of combinations for group 2) becomes:

Number of   Mathematical
Suit Bids       Expression
------------------------------
      1            COMBIN( 35, 1) * 21^0 = 35 * 1 = 35
      2            COMBIN( 35, 2) * 21^1 =  595 * 21 = 12,495
      3            COMBIN( 35, 3) * 21^2 = 6,545 * 441 = 2,886,345
      etc.
      35          COMBIN( 35, 35) * 21^34 = 9.025 E+44

Each of the above numbers is then multiplied by 4 for group 1) and then multiplied again by 7 for group 3). The result of all this generates the numbers that appear in the Stats table.


At this point we call in the computer. A simplified "C" program might look like:

Coef = 28.0;                             /*  Init coef with Group 1) times Group 3)    */
TotComb = 1.0;                           /*  Init count with "Pass, Pass, Pass, Pass"  */
                                         /*  For 1 through 35 "suit bids"              */
for (i = 1, j = 35; i <= 35; i++, j--) {
  Coef *= j;                             /*  Update the coef for the COMBIN()          */
  Coef /= i;                             /*  function.                                 */
  printf( "   %2d  %g\n", i, Coef);      /*  Output the next row in the result         */
  TotComb += Coef;                       /*  Update the grand total                    */
  Coef *= 21.0;                          /*  21 new intervening comb.                  */
}

Alternately a spreadsheet could be used (Will only require 35 rows).

And that's it. The output isn't formatted perfectly, but we'll leave that to the reader.


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