Sunday, September 23, 2007

The Odds and Ends of Bridge (Post Graduate Course)

Until poker became a TV star, everybody thought Poker was mostly bluff and luck and that duplicate bridge was a game of skill. Today we know that even the lowly internet poker players calculate the probability of finding an “out card” and live and die off pot odds. Conversely, club game bridge players largely consider bridge probabilities as an annoyance that only ruins an otherwise beautiful afternoon game. It can only be explained by the difference between money and masterpoints. If you don’t want me to ruin your beautiful afternoon, just exit this blog and go back to the novel. This is not easy stuff, but I hope that sometime it will form a useful reference. This is my last blog post on “Odds and Ends” and soon we will return to entertainment.

While bridge probability tables are helpful and avoid some difficult computations, the odds set forth in these tables are known as “a priori” odds, since they measure the probability of chances on any given hand prior to the play of any of the cards in that hand. As soon as one or more cards are played on the deal, these a priori odds are history. Sometimes the resulting change in the odds will be miniscule, but other times will be substantial. Probabilities calculated after the commencement of play are called “a posteriori” odds. These possibilities are too numerous to fully capture on any printed table.

The Deletion Principle

Let’s look at a simple example where we hold a seven card suit AKQxx opposite xx. If the suit splits 3-3 we will be able to take 5 tricks in this suit. When opponents hold 6 cards in a suit the “a priori” odds for the 3-3 split are 35%. The odds for a 4-2 split are 48%. With 6 cards outstanding there are 64 possible card divisions, 20 of them being 3-3 and 30 of them being 4-2 or 2-4 and the rest being nightmares.

Now, starting our analysis and play, assume that both defenders follow suit on the play of the Ace and King. We now know that the distribution of the suit was not 6-0 or 5-1 (our so-called “nightmares”). We have eliminated the 2 possibilities of the 6-0 break and the 12 possibilities of the 5-1 break, leaving only the 50 possible combinations of 3-3 and 4-2. By reducing the denominator, we have increased the odds of a 3-3 split from 35% to 42%, and the odds of a 4-2 split from 48% to 57%, but also note that the relative probability of the two suit divisions to each other has stayed the same. The probability of the 3-3 split is still less than 50%, and will remain less than 50% even if West follows suit on the third lead.

This example demonstrates an important principle of bridge probabilities known as the “deletion principle.” It can be stated as follows:

“When the opponents follow to the play of a suit with insignificant cards, the impossible distributions are deleted but the probabilities of the remainder of the distributions retain their relative proportionality to each other.”

Note that I have highlighted the words “insignificant cards.” You might at first think the holding of the J or Q by a defender would be significant. But in this example, the only division that will obtain 5 tricks is the 3-3 split, so the appearance of the Jack or Queen at anytime during the play of the suit would have no significance to the final result. If West held Jxx, and false carded the Jack on the second play in the suit, it would be an insignificant false card since we still have only one play that works for 5 tricks, the 3-3 split. We have no choice but to lay down the honors and hope that the jack was a false card. That is why all the opponents cards in this particular 7 card arrangement are considered insignificant.

Now let us change the cards a bit and demonstrate a holding in which the Jack does become a significant card and note how that changes probability of the 3-3 split. Assume we have AKQ10x opposite xx. Again we lead our two top honors with both opponents following suit with insignificant cards (not the jack!). As in the first example we now eliminate the fourteen 6-0 and 5-1 suit distributions, but there is another suit distribution that we can also eliminate in this case. We know that neither defender held Jx or we would have seen the Jack on the second round of play. Thus, we can also delete those 4-2 combinations that would have included a Jx doubleton by either opponent. Of the 30 possible 4-2 combinations, 10 of them involve Jx doubleton. While the probability of the 3-3 split remains unaffected at 35%, the probability of the 4-2 split has been reduced from 48% to 32%. Now in relative terms, the 3-3 split has a 52% probability and the 4-2 split a 48% probability.

Unlike the first example, we could delete the 4-2 combinations involving JX because a defender holding those cards will not make things easy for declarer by playing the Jack on the first or second lead if he held Jxx. Since now we know that the 3-3 suit split has become a slight favorite when our 7 card suit includes the AKQ10, we simply plunk down the queen and hope the odds work in our favor. The unwashed will sniffle that you got lucky, but we should let our non-readers suffer the consequences of not checking into Tommy’s Bridge Blog.

A10xxx opposite Kx. This example demonstrates that the favorable odds will also occur where there are two significant cards outstanding, the Q and J. The deletion process is a little different. Here we can delete the 4-2 combinations that include a Qx or Jx (18 cases out of 30) and the 3-3 combinations where either defender has both of the honors (8 cases out of 20). Again the combination of these deletions works out to a 52% probability in favor of the 3-3 split. The play is to ruff out a small card after taking the A and K, hoping to see the Q and J fall on the third lead. It’s a favorite! You might say, “Duh, how else would you play it?” Well the important thing is that you realize that you have now vaulted over that 50% threshold, and spiting the 7 card suit is now a preferable line of play to taking a finesse in another suit.

What you want to be on the lookout for in these situations are the combinations where opponents hold one or more significant cards and you don’t see any significant cards after the first two plays in the suit. What really makes an honor held by the opponents a significant card is that your 7 card suit holds a “threat card” (such as the 10 in my first example). This eliminates the possibility of a false card.

Here are examples of other layouts where the deletion principle enhances the probability of the 3-3 break to a 52% probability. The point being that you should try to split the 7 card suit, rather than take a finesse in another suit as an alternative. (i) 109xx opposite AKx. The play for 3 tricks in this suit is A,K and small to the 10. The 3-3 split is a 52% probability. (ii) A opposite KJxxxx. Play the A, K and ruff a small card if neither defender shows out. The 3-3 split is a 52% probability. (iii) AQxxxx opposite x. Here the 52% play is to play the Ace and ruff 2 small clubs trying to split the suit. It is also a 52% probability. It is not so obvious that the 6th card in one hand is the threat card, so memorize it!

The deletion principle is also the reason for the popular saying “8 ever, 9 never.” I won’t waltz you through the numbers, just keep repeating the old saying.

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