## Probibilities on opening hand at bridge

Hi all bridge lovers!

Bridge is a wonderful and very complex card playing game with thousands of books written about it.

For those who do not know it:
• Deck has 52 cards
• Rank per suit (high to low) is AKQJT98765432
• Points per card: A=4, K=3, Q=2, J=1, rest = 0
On each deal, every player gets 13 random cards. The game starts with a bidding phase, ending for instance in "NS 4H", meaning that North/South aim to make 10 (out of the 13) tricks with H(earts) as trumpsuit.

Now, to get a higher level of expertise at the game, I must gain more knowledge about the probilities that are important in this game, for instance, given a certain distribution of spades between North and South, what is the best way (an optimum of "max tricks" versus "max probility") to play the suit.
There are some books that specificcally address this problem, with a well described mathematical background (combinations/permutations, etc) on how to calculate best results. These books deal with situations where a distribution whithin a partnership is know, for instance:
• North: AJT76
• South: Q98
In this case, only the K is missing, and it is very easy to calculate best chances of how to play for maximum (or garuanteed) number of tricks in the suit.

However, bridge starts with the bidding phase as I mentioned, and to my knowledge, there are NO books that deal with hand evaluation bidding strategies (should I bid, and if so, what to bid) in combination with a solid mathematical foundation.

The bidding phase starts with the first player trying to evaluate the "trick potential worth" of his own 13 cards. This is normally a combination of counting points (number of AKQJ, 25+ is considered extra worth, due to the reward system) and distribution (8-card is worth a lot more than 2-card, easier to make more tricks).

Extreme examples:
• North has all 13 spades? --> 13 tricks in a spade contract (with only 10 points)
North has 28 points (all aces and kings), but only has 8 tricks
Of course, many books have been written how to deal with that, but none of them reveal the "WHY" in a mathematical sense.

What I want to do, is devellop a computer program that tell's me exactly "WHY".

And to not let things get too complicated, let's start with the evaluation "per suit".

Examples, per suit, for the first bidding player (North), with no other information:
• North has: AKQJT --> 100% certain, 5 tricks worth
• North has: AQJT9 --> How much % exactly?
• If partner South has K (33%) --> 100%
• If East has K (33%) than still make 5 tricks if K singleton, else 4 tricks
• If West has K (33%) than it depends on the number of cards South has in the suit
• If South has 0, E makes a trick on 2+card
• If South has 1, E makes a trick on 3+card, etc