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Monty Hall, Restricted Choice & Vacant Spaces
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Bridge players are particularly fond of the Monty Hall query because it provides a prime illustration of the principle of Restricted Choice (RC). When the coveted prize has been placed in the selected box, MH has a choice between the other two; when the selected box is empty MH has no choice, he must open another empty box. According to the RC we go for the case where MH has had no choice, switching must be twice as promising as staying put. It must be a compelling argument for a bridge player.

RC principle provides a powerful shortcut − it enables the declarer to identify the best line of tackling the remaining of a suit on the basis of the information from the cards of that suits played so far by the opponents, with no need of any involved calculations.

Here is an intricate application of the principle. South in 3NT, with Kxx in a suit opposite 108x in dummy, needs to cope with a honor, the queen or the jack, from East. With only one honor in his hand East has no choice, with both he can play any of the two cards. According to the RC, declarer is advised to play the king, and after it has been captured by the ace, to cover West’s return with the ten in dummy.

The first play is obvious, and the second is the consequence of the first. The nine is still equally likely on both sides, while the remaining lower honor is twice as likely in West as in East. Moreover, the double RC allows you to correctly predict the success rate of 80% of the combinations of A, Q, J and 9 where declarer play matters. Indeed, South can’t go wrong against QJ in West - 4 cases, nor AQJ in East – 2 cases. Out of the remaining 10 cases, the proposed line loses only 2 cases, that of QJ and QJ9 in East, and wins the other 8 cases.

The Vacant Spaces (VS) principle is another useful tool in the hands of a skilled declarer. The main difference is that the VS depends on the information not from the suit itself but from the information about other suits. Its execution is straightforward provided that only a single card needs to be located and that the information about other suits can be presented in a simple form. It can be applied also in more involved cases but the calculations won’t be as simple as a comparison of the numbers of vacant spaces.

An often quoted problem is that of handling the trump suit, let us say spades, consisting of AKQ10xx opposite xx in dummy, after a heart preempt by the LHO. When both opponents contribute small cards to the first round and a half of spades, should declarer finesse the jack or rather play from the top. The relevant distributions of the suit are Jxx – xx, Jx – xxx, and x – Jxxx. Thus, it isn’t only the jack that matters, the unseen small card is important as well. Can the count of vacant spaces be of any use here as well?

Let me tell you that the heart suit is known to be divided 7 in West to 2 in East. We begin by estimating the odds for individual divisions of the missing spades. What are the odds for J2 – 543 against 2 – J543? These two holdings differ only by one card, the jack. Counting the hearts and the spade pips we have 8 known cards on the left and only 5 cards on the right. Accordingly, the odds for the jack in East versus jack in West are like 5 to 8. For both splits of the missing spades with Jx and x in West we have four allocations of the small cards, the odds for the these two distributions remain 5 to 8.

What about J32 – 54 against J2 – 543? The difference is again only one card, and this time we get the odds of 4 to 9. However, in the former case we have six possible allocations of the low cards, in the latter case only four. Adjusting for this difference we get for Jxx – xx against Jx – xxx the odds of 2 to 3.

Now we need to put the two pairs of odds next to each other. The common distribution in both comparisons is that of Jx – xxx. With the common denominator of 15 in the middle the odds for the three relevant distributions Jxx : Jx : x in West are like 10 : 15 : 24. By taking two numbers on the left together we get 25 for the jack in west against 24 for the long jack in East. Would you have guessed that going from the top is theoretically better than finessing? But only just. And perhaps not quite. Why not?

The point is that I didn’t reveal the whole information available to the declarer at the table. The purpose of the exercise was after all to demonstrate how the compound odds can be calculated without combinatorics, statistics and electronic devices. At the table? Yes, with some practice also at the table. However, I don’t recommend spending your precious time on such futile exercises. Why not? For a number of reasons.

An experienced declarer should be able to identify the most promising line without much ado when the difference is sizable. Otherwise, when she is in a serious doubt, the difference is apparently small, hence insignificant. I gave you nothing more as not to distract you from the purpose at hand, and not make the calculations intractable. Yet, there are plenty of other aspects of the deal worth of your attention.

On a real deal your estimate of the likelihood of a stiff spade in East will depend on any of the following: Did East bid 3 or 4? Was he vulnerable? What did he lead? How fast? All of that taken together is apt to put more weight on your decision than arid probabilities calculated skillfully when putting all practical aspects of the deal aside.

Let me finally show to you how easily Phil Martin, a renown sleuth of probabilities with some 30 years of experience, can lead you astray when leaving aside a decisive piece of information (nothing personal Phil.) Phil ends his 1989 contribution to Bridge Today, still to be found on his website, with the following dummy play problem.


West leads the deuce of clubs (third and fifth) − low, nine, ace. You play another club. East wins with the queen and leads the king of clubs to tap dummy. West follows to all three clubs. What is the percentage play in spades?

You cash the spade ace and continue with a small one. West having shown 5 black cards has 8 vacant spaces, while East having implied a 5-card club suit and having followed to the first spade trick has 7 vacant spaces. The simple VS argument points toward the queen in West. Not so fast, says Phil. East having opened 1 has denied a 5-card long red suit. On his reckoning this counts for a half card for each suit, to be added to the vacant spaces total, leaving 8 effective vacant spaces for East as well. Phil concludes that “it's a toss-up between the two plays. Both the finesse against West and the drop are about sixty percent.”

Hold on, Phil. You have just ignored two essential considerations. First, for a fighting chance in your spade game you need the diamond king in East, which will cost one vacant place there. Hence with all due corrections you will end with the original count of 7 virtual vacant spaces in East. I did the required calculations the hard way, and came to the odds of 35 : 33 in favor of the finesse. Not much in it, you could say. Is that all?

Not yet, I have an argument at the bottom, which will make all calculations redundant. So far we have overlooked what hasn’t happened, namely the fact that East didn’t open the bidding with 1NT. That ends the case in one stroke. With balanced hands in the appropriate range, including Qx KQ109x East would open with 1NT rather than 1. There will be many such hands in particular for defenders employing weak no trumps. All these excluded, the spade finesse wins with flying colors. It isn’t even close.

I’ll leave it here, with still some crumbs left for the connoisseurs of this famous couple, the RC and the VS. My own concern as declarer when mulling over a hand is not that I would miscalculate chances, but rather that I could have overlooked an essential piece of information.

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