A Look at the Odds


I'm not an odds maven and have little use for the type of analyst who tells us one line of play was 3/128ths better than another, which I saw in one column. I doubt if many players would have the capability of figuring out such a difference during play, and doubt if many would want to tie their game to such mathematical gymnastics. So this is directed to the casual player who wants to improve without sacrificing the fun.
There are two maxims about the odds that I have often cited. The first is that an odd number of cards will tend to split as evenly as possible, while an even number of cards will tend to split a little unevenly. Thus 7 cards are favored to split 4-3 (62%), while 6 cards are favored to split 4-2 (48%), the 3-3 split following in likelihood (35%). You would find much the same in other odd and even numbers of cards, but I think you'll often find those odds just cited among the most common you'll have some interest in. And the second maxim is that though you may feel yourself too free-spirited a player to get bogged down in mathematics, there is a point where a cavalier disregard of the odds sinks into bad bridge.
Here is another misperception about what the odds are all about: "I followed in 87% chance and went down while that li'l ol' lady over there went onto a poor percentage play, a 13% chance, for God's sake, and wrapped up the contract. Jesus!" I didn't say a word (and this was a person who prided himself on being able to elucidate any topic in bridge). But I thought plenty. I wanted to ask, "Do you think 87% chances should work for you 100% of the time or 87% of the time?"
Some months later, there was another rather similar take on the odds. It was a Saturday afternoon, and I believe there were two or three kibitzers around the table (myself being one of 'em) when a declarer in four spades got every piece of bad luck you could imagine. Cheeze, was the general consensus around the table. The hearts have to split 4-2, the queen has to be in the short hand, the long spades have to lie with that defender, the A of clubs cannot be in the hand with the long hearts, and on and on. I'm making up the stipulations here for I don't remember 'em, but they were in that order: four, five, six things that had to go wrong for declarer to be unable to bring the contract home. And when they were done, I offered the observation that "That must have been about a 98% game." At which the defeated declarer looked at me with a beady eye and said, "Oh, yeah? Then how come I didn't make it?"
I don't recall if I said anything or if someone jumped in, but I know I didn't say what I was thinking or at least am thinking now, which is: well, is 98% the same as 100% or are they two different figures? And when you get clear on that, you'll know why this hand went down.
So this is the way of the odds. They don't promise anything on any one hand, unless you've got a 100% chance, of course. And this is why tournaments aren't decided in two or three boards, nor are the big ones decided in 26 nor in 52 boards. So you've got to figure that, everything else being equal, in the long run you're going to come out ahead of the opponents if you pay some attention to the odds. If you go against the odds, sure you'll come out smelling like a rose from time to time. But lemme ask one question: Do you want to put a smile on your partner's face 68% of the time or 33% of the time? I think you'd do well to avoid flagrant challenges to what the odds suggest both to bring a contract home more often than you will with a disregard of the odds and for the respect you might engender in both your partner and your opponents.
Here's another twist to the odds:
A Q 5 4
A 3
A J 9 8 2
A 8
. . .. . .. . .
K 10 9 8 3
J 10 9 8 2
------ Opening lead: Q of clubs
K 6 5 Contract: 6 spades Vul: Both

What are the odds that clubs are splitting 7-1? Well, I go to my trusty mathematical table in the ACBL encyclopedia and find that the odds of such a split are . . . 2.86%. Minuscule, eh? But I say the odds are about 95%. Ninety-five percent! What am I, crazy or something? That's way out of bounds, huh? Oh, wait a minute. Did I forget to give the auction. It went like this:

EastSouthWestNorth
3 Dbl 4 Dbl
Pass 4 Pass 6
All pass

Ah, that's different, huh? An opponent makes a vulnerable 3-level pre-empt with at best a jack-high suit and you wonder if that lead was a singleton? Oh, you don't have to tell me that some people will make that outrageous bid on a six-card suit. I didn't say the odds were 100% did I? But you've gotta suspect or surmise that East more than likely has a 7-card suit, leaving just one for West. Here is the whole hand.
A Q 5 4
A 3
A J 9 8 2
A 8
J 7 6 2 ------
K Q 7 4 6 5
K Q 6 4 10 7 5 3
Q J 10 9 7 4 3 2
K 10 9 8 3
J 10 9 8 2
------ Opening lead: Q of clubs
K 6 5 Contract: 6 spades

This was an actual case, incidentally. Declarer won the lead in dummy and promptly led a club to the king and got ruffed. And did he thus go down? No, it remains to be told that West now led a low heart for totally unknown reasons when he can see that declarer has a heart loser on the lead of the K. I don't know why. But that part is incidental. Declarer ignored some information that should have warned him against a second round of clubs and should have been defeated.
[In retrospect, a few years later, it looks to me as though declarer would've had a tough time with that 4-0 trump split even had he not coughed up that ruff by West. But that's neither here nor there. The point is that though the odds are reliable enough -- in the long run -- when you have no other information, there are times where the bidding or play gives you more information than the odds could, such as that pre-emptive bid above.]
So I say all side-suit leads in a trump contract must be viewed with some suspicion. The odds are geared to telling you what happens in a hundred or a thousand hands, not on this hand. You're missing five diamonds to the queen in an unbid suit? And a low diamond is the opening lead? I would view it as a possible singleton even though the odds don't favor a 4-1 split, of course. That's not to say you should never try the second round in the suit in such a circumstance. Who can predict what vagaries will arise in the cards? But it is to say you'd do well to avoid that suit before trump are out if feasible. The odds don't tell about the distribution of the cards on this hand but only on every hundred hands, and you would do well to pay some heed to what the bidding, signalling and plays of the opposition tell you as much as you figure out the odds. The odds, indeed, are for when you don't had much input from the opponents. Then you have nothing better than the odds.

I have a column by a professional columnist who tells us that some readers thought looking to a 2% better chance was silly. He then went on to show how a player had improved his chances by 2% to make his slam. My first thought was that "silly" wasn't quite the right word, but on reflection, I decided that it was. That is to say, for the 99% of bridge players who have no pretensions toward turning professional. For those who make a living out of playing bridge and may average, say, 30 hands a day, I say, be my guest. For them to get down to the nitty-gritty of working out the odds and develop the talent to do so quickly at the table is entirely understandable.
But for the great, great majority of bridge players, I have two reasons for agreeing with the word "silly". One is that it's very doubtful that many bridge players have either the inclination or the mathematical skills to work out the odds on a hand to that fine a point. I might suggest that a 2% better chance is going to benefit you only 2% of the time -- when you get that hand where there is a 2% difference between two lines of play! Which is to say (by my reasoning) once every blue moon. The counter argument, of course, is that the skill to work out a 2% difference will work for 3% and 4% differences, etc., which will occur more often than once a blue moon. Granted. And I already allowed that it's reasonable for professionals to develop that skill down to 2% differences. But I think the people who have a general grasp of what the odds favor are going to enjoy themselves a lot more not trying to work out a 2% difference at the table.
But I have another reason for more or less scorning that drive toward looking to a 2% advantage, and this one I actually feel more strongly about. And that is that this simply cannot rank very high in the hierarchy of butchered contracts. Mind you now, I didn't say disregard of "the odds" cannot (and I have given two maxims above that even novices should be aware of) but that disregard of relatively small differences, call it under 10%, cannot. I would probably rank that about 25th if that high.
I have elsewhere listed the reasons (as I see them) for butchered contracts, starting with a lack of counting. That's counting involving no higher level than second-grade arithmetic. The second reason I have given is a failure to take natural finesses, which of course is a flagrant disregard of the odds, but of an obvious disparity in odds which I have in no way scorned as taking into account. Beyond that, I think of a failure to draw trump, a failure to see the advantages of a dummy reversal, especially among those who through a transfer bid deliberately put the long trump suit in dummy. That's a bigger reason than these small difference in odds. Further, I have often told the reader, don't cash out your established winners without good reason until you're ready to cash out. That is a marked tendency (apparently in the hope of inducing a stupid discard from the defense), when as I have pointed out, if it's a balanced suit, you'll very likely be wiping out entries you could use later, and if unbalanced, you'll be as likely as the defense to find yourself discarding useful cards. I could probably give 25 failures to play a dummy reversal for every 2% advantage you can dredge up, or an equal number of contracts butchered because one simply didn't bother to draw all trump. Or to go a bit further, I could give a hundred hands of only those reasons just given for every 2% advantage you can find. A hundred. And that's why I agree with the word "silly".


For those who have had enough: Some Illustrations
Not tired yet?
Some more thoughts on the odds.