About a Girl Named Florida

No sooner had I finished reading Mlodinow’s book, than I received a call from a very nice lady who had decided to take up bridge upon retiring from a job in which she had dealt constantly with statistical data. Prompted by her love of numbers, she was drawn to my book. I congratulated her on her choices and wished her years of happy entertainment without mentioning the frustrations that go along with the game. However, the kind lady had called not to praise my book, but to correct it. She was familiar with The Monty Hall Problem, and was convinced that my treatment was wrong. I apologized for my inadequate explanation of the process, but happily could refer her to ‘The Drunkard’s Walk’, for a fuller treatment of the problem and its resolution, and thus for an independent confirmation of the validity of my approach. I hope she followed my advice, recovered my book from the trash bin, and corrected her long-held views.

The Monty Hall Problem is one often used by bridge writers to illustrate the application of conditional probability to card play, in particular, through the Principle of Restricted Choice. Now Mlodinow has provided us with another illustrative example of the Bayes’ Theorem at work that demonstrates directly the difficulty many encounter with the concept of probability linked to the current state of partial knowledge. He calls it ‘The Girl Named Florida Problem’. A detailed description may help in seeing the connection.

Suppose that a couple have produced 2 naturally conceived children. What are the chances they are both girls? We assume that at the time of conception a boy is as likely to result as a girl. The event is mathematically equivalent to tossing a coin. First we present a false argument that was common in centuries past which goes as follows. There are 3 equally probable states: 2 boys, 2 girls and a boy and a girl. The chance of producing 2 girls is 1 in 3? That is wrong because the probability of a given outcome of a series of random events is proportional to the number of ways in which that outcome could have been produced. One must take into account the birth orders, of which there are 4: boy-boy, boy-girl, girl-boy, and girl-girl. The chance of producing 2 girls is 1 in 4.

Pascal would have got it right, as would most bridge players who are asked, given that 2 finesses are to be taken, what are the chances both succeed? There are 4 equally likely possible outcomes of the play, for two of which one finesse wins and the other loses. The chance of both finesses succeeding is 1 in 4.

Next we ask if one child is known to be a girl what are the chances the other is also a girl? It would be wrong to argue that given that one is a girl doesn’t affect the odds the other is a boy, so the chances of 2 girls should be 50%. The correct argument is that the birth order of boy-boy has been removed from consideration, so there are 3 possible sequences remaining leaving the chances of 2 girls at 1 in 3. Similarly, if we are assured that at least one finesse wins, the chance of the other also succeeding is 1 in 3.

Next we ask, what are the chances of 2 girls given one of them is named Florida? There are those who would argue that whether the girl was named Florida, or Jane, or Laura should make no difference to the odds that their other child is a boy. Although the name Florida is unusual, there is no causal effect at work. Consider the problem statistically and imagine going through US census data looking for all parents with 2 children one of whom is named Florida. Can we expect to find that the other child is a more likely also to be a girl? It doesn’t make sense that we should.

Although there seems to be no causal link between the name and the probability of 2 girls, the argument doesn’t solve the Girl-Florida problem as posed. The correct solution is obtained by incorporating the information that a daughter is named Florida, condition FL, into the possible sequences of births. The possibilities are the following 4: boy- FL, FL-boy, FL-girl, and girl-FL, in half of which the other child is a girl. This corresponds to our intuitive feeling that it doesn’t matter whether the girl was named Florida, or Jane, or Laura, the chances are 50-50 the other child is also a girl. What does matter is that the naming of one child changes the odds for a second girl from 1 out of 3 to 1 out of 2.

The application to bridge probability is straightforward. Suppose the opponents hold the ♣ 4, ♣ 3, and ♣ 2. If we lead the ♣ A from hand and the LHO follows with a low club, it may not matter whether the played card is the ♣2, ♣ 3, or ♣ 4, but what does matter is that the card is specified. There is a difference between ‘a low club’ and the ♣ 2 being played, just as there is a difference between ‘a girl’ and ‘the girl named Florida’, the difference being in the amount of information being made available. As birth order must be taken into account in the 2-child family problem, so the order of play must be taken into account when following in a suit. If the LHO is seen to have been dealt the ♣ 2, all the possible combinations for which the RHO has that card have been eliminated.

What is the connection to The Monty Hall Problem? The formulation is the same. One begins with a set of conditions of known probabilities. Information is provided that reduces the number of possibilities. In The Monty Hall Problem, a door is opened to show that the prize does not sit behind that door. The probabilities are re-evaluated on the basis of the remaining conditions. These conditional probabilities must add to 1, as only these possibilities remain. Mathematically the process is expressed by Bayes’ Theorem. One must be careful when describing the process in normal language which is not well suited to describing random activity. Hence the need for books that attempt to get around this fundamental difficulty that affects even the most intelligent readers.

Doubling in the Dark

We now turn to the problem of finding patterns in data with random components. It is dangerous to draw an unbiased conclusion as our upbringing leads us to seek out examples that justify a prejudiced view. When I saw that the Canadian National Open Team Finals was to be a match between a team with two Big Club pairs and a team with two 2/1 pairs, I expected the Big Clubbers to triumph, as they did. The interesting question is this: to what extent can it be claimed that their victory was due to the use of a superior bidding system? We have space only for a few disappointing counter-examples.

One advantage of Precision’s limited bid strategy is that the user has a better chance of controlling the flow of information. If partner opens with a Big Club, every effort can be made to extract information concerning conditions that might constitute a rare set of circumstances in which a slam is worth bidding. If partner has opened with a limited bid, one may cut off the flow of information by jumping to game with substantial values when there is little perceived need to search for alternative contracts. That is the scientific explanation. There is also a psychological as well as technical advantage to getting in first. By opening light one may inhibit the opposition’s constructive bidding. If that is your style, partner has to carry on as if your bid were normal. This may work well if the illusion created matches reality in critical aspects, such as suit quality.

Watching the Finals on BBO, I was reminded of a one-act play I attended in a London’s West End theatre some 4 decades ago. The curtain went up on a stage in darkness, but we heard the actors carrying on the polite, boring conversation of one upper-middle-class couple visiting another in their London flat in the evening. After a minute or so of that, just as the audience was growing restless, the stage lights came on, and one of the actors exclaimed, ‘Oh, damn, there goes the electricity again!’ After that things got amusing as we could watch the couples moving about in the dark, bumping into each other, doing things they knew the others couldn’t see, their conversations never quite matching their actions. Much laughter, but eventually the ideas ran out, the lights came on (out, that is) and the conversation returned to normal as the couples bade their conventional farewells.

As I say, it was a one-act play, and more of the same would have become tedious. I felt the same after watching the finalists play blind-man’s bluff for 32 boards over a scheduled 128 boards. The auctions never seemed quite to match their holdings, and bluffing was a major strategy in view. Here is a deal that gives the flavor of the contest.

Board 22

Dealer: East Vul: E/W	North ♠ A 3 ♥ A K 9 7 2 ♦ K 10 7 ♣ A 9 4
West ♠ Q 10 7 ♥ Q 10 4 ♦ 9 8 ♣ Q 10 6 5 3		East ♠ K J 9 8 6 5 2 ♥ J 6 5 3 ♦ A J ♣ —
	South ♠ 4 ♥ 8 ♦ Q 6 5 4 3 2 ♣ K J 8 7 2

 

Wolpert	Campbell	Korbel	Klimowicz
—	—	3 ♠	Pass
Pass	Dbl	Pass	4 NT
Pass	5 NT	Pass	6 ♦
Pass	Pass	Dbl	All Pass	Down 1 for -100

 

Daniel Korbel had an intriguing problem as to what call to make on a hand with 7 spades, 10 HCP, a void, and 6 losers. There are very few who would pass and await developments, so if one feels compelled to bid, 3♠ is reasonable, as partner can read you for a pretty good hand; it combines preemption with a bare modicum of construction.

Ross Taylor wanted to get out to the best minor suit game, but Gordon Campbell liked his hand a lot – 8 controls, the stuff that slams are made of. The odd aspect of 6♦ * was that it was the preemptor who did the doubling. Good move: Korbel wanted a club lead, but he didn’t get it. No problem this time, as Darren Wolpert had enough weight to hold it to 11 tricks. One could say that the auction was typical of thousands one might encounter at a local club duplicate – nothing really wrong, but nothing really right.

Judy and Nicholas Gartaganis have represented Canada internationally with great success. They play their idiosyncratic form of Precision to the hilt. Superficially their bids appear normal, but often the effect is to create an illusion. Theirs is a style where the offensive potential of the hand from the bidder’s perspective is taken into account, so their bids are more judgmental than descriptive. If this puts off opponents used to more conventional evaluations, so much the better. Here Judy G. saw her hand as a 1♠ opening bid. Points! Schmoints! Add 5 points for a void, and you are at the top of the range for a Precision opening bid (joke). Unfortunately, there is always a partner there to mess it up, as follows.

 

Nick G.	Balcombe	Judy G.	Taylor
—	—	1 ♠	Pass
2 ♠	Dbl	4 ♠	4 NT
Pass	5 ♣	Pass	Pass
Dbl	Pass	5 ♠	Pass
Pass	Dbl	All Pass	Down 2 for -300

 

When Keith Balcombe bid 5♣ he was already too high, and Nick G. could guess that his clubs meant trouble, however, there appears little need for a double as the profit might not be greatly increased by that action. 4♠ was not likely to make. He could have passed and gained 4 IMPs on the board, but that is not his style, so he doubled for maximum profit. Of course, Judy G. had to pull the double, losing 12 IMPs on the board.

The above deal provides a lesson on how to compete against flightly bidders. Be more concerned about finding out what you and your partner have rather than what the opponents say they have. Bid your cards to the hilt and turn uncertainty to your own advantage. The guessing game works both ways, and it is unlikely the light bidders can double with assurance. On this hand Ross Taylor made a proper 4NT takeout, and Keith Balcombe, with both opponents getting into the bidding, was not tempted to go for slam. He was happy to double and take his profit. Having bid his values, Taylor could pass.

In the Land of Wishful Bidding

If the cards cooperate, it pays to direct one’s bidding towards a particular end from the start without attempting to deliver full disclosure. Under this scheme the minors take a back seat. After all, with this hand: ♠54 ♥Q8 ♦AK8752 ♣AK9, would you prefer to play in 3♦ making 110 or 3NT making 400? Rather obvious, even if the chance of making 3NT is only 1 out of 4. Mathematically close, but in a long match you’d still opt for 3NT, as there may be psychological advantages to bidding and making a bad 3NT.

What is your bid with the above hand if your RHO opens 1♣ ? Rather than introduce diamonds in the hope partner will be able to bid something useful, why not bid 1NT, suggesting 3NT? No one will guess you have such a potential source of tricks, but partner may be able to make an encouraging move showing values in the majors. Indeed, that is what happened but the end result left something to be desired, when partner took the 1NT bid at face value and, not surprisingly, attempted to declare the hand in a major suit.

Board 16

Dealer: West Vul: EW	North ♠ 5 4 ♥ Q 8 ♦ A K 8 7 5 2 ♣ A K 9
West ♠ A Q J 5 ♥ A 7 5 4 ♦ J 3 ♣ J 7 4		East ♠ 7 3 ♥ K J 6 ♦ 9 4 ♣ Q 10 6 5 3 2
	South ♠ K 10 9 6 2 ♥ J 9 3 2 ♦ Q 10 6 ♣ 8

 

Wolpert	Nick G.	Korbel	Judy G.
1 ♣	1 NT	3 ♣	4 ♣
Pass	4 ♦	Pass	4 ♠
Dbl	5 ♦	Dbl	All Pass	Down 2 for -300

 

At the other table the doubling was in the dark. Campbell could open a weak NT, and Balcombe doubled to show values. Klimowicz managed an escape to a safe location, which Taylor doubled, he thought, for takeout. Balcombe could have gained 9 IMPs merely by bidding his 6-card suit, but greedily he passed and held the gain to 3 IMPs. This shows once more why successful doubling of partials has become a lost art. One must first bid one’s values, and if the opponents carry on further, then double and let an informed partner decide the issue.

 

Campbell	Balcombe	Klimowicz	Taylor
1 NT	Dbl	Pass*	Pass
Redbl*	Pass	2 ♣ *	Dbl (Takeout?)
All Pass		2 ♣ * making for	+180. Ugh!