When this hand arose in the European Championships in the 1950s, the British declarer did well to play five diamonds rather than three no-trump. The defense cashed two rounds of hearts and played a club.

Naturally, declarer took this and led a top trump, on which West played the eight and East the three; then declarer took both top spades. When he led the diamond jack, West followed suit. Now South had to decide whether to play for spades to split and diamonds not to behave (when the right play would be to duck in dummy and draw a third round of trumps), or for the diamonds to be 2-2 and spades to be 4-2 (in which case South should overtake the second trump and would then be able to ruff out the spades).

Spades are slightly favored to break. But how likely is it that diamonds break? Imagine West has the diamond 10-9-8 and East the three, compared to that suit splitting 2-2? In abstract, the individual singleton is less likely than any individual 2-2 split, but in addition there are three doubleton honor-pairs where West must follow with his two cards at his first opportunity, and East similarly has to play his three at his first chance, lest you overtake on the second round with impunity.

So it is clearly right to play to overtake the second diamond, as that suit is much more likely to split than spades. The fact that this was the winning line does not prove anything, but at least virtue was rewarded.

All our deals this week have focused on the principle of Restricted Choice; but this concept cannot be considered purely in abstract, as today’s deal from the later stages of a Vanderbilt Trophy match shows.

Declaring four spades, your chances do not seem so great on a devious low diamond lead, but when the diamond king holds the first trick, things look up. Draw two rounds of trumps, ending in hand, and lead a heart to the jack and king. Now East cashes the diamond queen and returns the heart 10, on which West unblocks the queen.

You ruff the last heart in hand, cross to dummy in trumps, and lead a low club, intending to put in the seven. Naturally, East thwarts you by playing the nine, so you try the queen, losing to the king. When West returns a low club, what should you do?

If you have been following this week’s theme, you may conclude that Restricted Choice suggests playing low. The logic for that would be that East is more likely to have J-8-x or J-9-x than 9-8-x. However, there is a much sounder argument for putting in the six, if you remember the earlier play. East can be assumed to be a true expert player; with three low clubs, as opposed to jack-third, he would not have defended this way when on lead earlier in the hand. He would have shifted to a club rather than returning a heart, to break up the impending endplay.

The Monty Hall problem arose on “Let’s Make a Deal.” The contestant was presented with three doors; behind one was a car, and behind the other two were goats. The contestant selected a door, then the quizmaster opened a different door to reveal a goat. Now should the contestant change his selection to the third door or stick with his original choice?

Interestingly, the contestant doubles his odds by switching his choice. He’d turn his car into a goat only if he had correctly selected the car with his first pick — a 1-in-3 chance. But he would win the car if he had originally picked either of the two goats — a 2-in-3 chance. The reason comes back to Restricted Choice; in each of the latter two cases, the host had only one goat left to reveal. However, if the first pick was the car, the host had a choice of goats to show you.

The biggest caveat for Restricted Choice comes when a player contributes a significant honor or spot-card where that play is not forced. Consider today’s slam, where when declarer cashes the heart king, he sees the 10 fall from East. The singleton 10 might seem more likely than J-10 doubleton, since we must reduce the chance of the latter by half — because East might have followed with the jack from that holding.

That is true, but East wasn’t forced to play the 10 from J-10-7, his actual holding in the diagram. Maybe only an expert would be capable of that false-card, but that is a completely different issue.

This week we are addressing the thorny problem of Restricted Choice in bridge. This dictum says that if (and only if) a player had a choice of equal cards to play, then the probability that he had one of those cards singleton should be compared to half the probability that he had both of those cards — because with the doubleton he might have played the other card. Granted, this does assume that he was equally likely to play the queen or jack from queen-jack doubleton, but unless you know to the contrary, you should indeed assume that.

A deal may make the point more clearly than words. Declaring four hearts, you win the diamond lead for fear of a club shift, after which the defenders might maneuver a ruff. You cash the heart ace, dropping the 10 from East. Should you now lead a heart to the nine or to the queen?

As indicated above, it is correct to play East for the doubleton K-10 rather than for the J-10 doubleton. That is because, with the former holding, he had no choice but to play the 10 at his first turn, whereas with the J-10 doubleton he might have played either of those cards. Thus, one should not compare the initial probabilities of each doubleton holding, which are equally likely, but instead assume that the K-10 is twice as likely as J-10 doubleton.

How does that relate to the Monty Hall problem? We will find out tomorrow.

Yesterday, we mentioned Occam’s Razor, a hypothesis dating from medieval times. It states that when comparing two explanations, one should assume the truth of the one with the fewer assumptions. This applies to bridge in the form of the Theory of Restricted Choice — and, as we shall see later this week, to what is popularly known as the Monty Hall Problem.

In bridge terms, when comparing two possibilities, we must reduce the probability of an event if a player had previously had a choice of equals to play; this is because he might have played either of them at that turn. But enough theory — let’s look at a deal and see how it works in practice.

In three no-trump, you win the spade lead and drive out the heart ace, then win the spade return and cash the hearts, both defenders pitching small diamonds. With no clue as to who has the fifth spade, you need to bring in the clubs now.

You cash the club ace, then cross to the club queen, bringing down the 10 from West. Should you finesse or play for the drop on the third round? The appropriate percentages to measure up are jack-fourth or 10-fourth of clubs in East against J-10-x in West. You should not look at just the chance of jack-fourth against J-10-x (where the odds would be very close), because West would have had a choice of high spot-cards to play from that holding at his second turn. That makes the finesse the clearly indicated play.

Sixty years ago, Terence Reese produced his seminal work, “The Expert Game,” published in the U.S. as “Master Play.” This book introduced a variety of plays that are now part of every top player’s armory. And the most important idea that the book promulgated was the Theory of Restricted Choice.

This theory borrows from William of Occam, who invented Occam’s Razor. This states that when faced with a choice of competing hypotheses, one should select the simpler option.

How does this apply in bridge terms? Consider today’s deal, where against your contract of five clubs, West leads the heart jack. East takes his ace and returns the suit, letting you ruff. How should you play the trump suit?

Clearly, East is more likely to be short in clubs than West, not only because East has the long hearts, but also because if East had three clubs, he might have been able to shift to a singleton in spades or diamonds at trick two. So lead out the club king; when East follows with the jack, you play a second trump, West following with two small cards, leaving you to decide whether to finesse or play for the drop.

The percentages here might be misleading: A singleton jack is less likely than the doubleton queen-jack, but if East had doubleton honors, he might have followed with either card. So the true percentages to compare are the singleton honor against half the percentage associated with Q-J doubleton. Playing for the finesse is therefore clearly right.

Declarer might easily have relaxed in six no-trump here when West led the heart 10. There were 11 top tricks, with 13 tricks available on a 3-2 spade break, and 12 tricks even on a 4-1 break.

But declarer carefully won the first trick with the heart ace, then carefully cashed the club ace-king followed by the heart king. It was only then that he led a low spade to dummy’s queen. West’s discard turned a potential 13 tricks into 11.

South continued with a low diamond from the table and took the jack with the king, then ran the diamond 10 to East’s queen. East exited with the spade jack, taken by dummy’s ace. Now declarer cashed the diamond ace, pitching the heart queen from hand. He followed up with the heart jack and discarded the club queen from hand, bringing everyone down to three cards.

East was reduced to the spade 10-9 and the diamond nine, and when declarer called for dummy’s club jack, East had no winning discard. He threw the diamond nine, and now dummy’s eight was high.

If East had followed low to the first diamond, declarer would have put in the 10. Had this lost to West, then on any return, declarer would have cashed the heart, diamond and club winners, then played a spade to the ace to cash the diamond ace. This would execute a simple spade-diamond squeeze whenever East had started with four diamonds. It would also work fine when East had begun with at most three diamonds including the nine and at least one diamond honor.

Liam Milne was able to report a fine play by Barbara Travis from the semifinals of this year’s Australian Women’s Playoff. Travis declared six spades on the friendly lead of the club eight to the jack and queen.

If clubs behaved, 12 tricks would be easy; but West’s decision to lead a club instead of a heart argued strongly that the lead was from shortage — and that West must not have the heart ace, or a club lead would be almost pointless.

With the general idea of playing East for the heart ace, Travis ruffed a diamond in the short trump hand, then drew four rounds of trumps. When West followed all the way, she provided additional weight to the theory of club shortness in that hand. Dummy discarded a heart and a club, while East discarded a low heart, a low club and the diamond jack.

Travis now played the last trump and the diamond king, coming down to three clubs and the bare heart king in dummy. East, holding the doubleton heart ace and three clubs, had no choice but to come down to the bare heart ace. Trusting her judgment, Travis cashed the club 10 before exiting with a heart. East had to win and, with only clubs left, was forced to bring dummy back to life. From declarer’s perspective, the only thing that could have made this hand any more spectacular would have been if both heart honors were off-side!

In the Seniors, Open and Women’s events, most declarers who received a club lead reduced themselves to guessing hearts — and not all of them did so correctly.