When it comes to restricted choice, all books on declarer play offer the same reassurance: a second finesse of dummy's AJT tenace has roughly a 2:1 chance to succeed. RHO is 50:50 to falsecard from the KQ the first time around, so the fact that he shows you the queen reduces the probability that he holds the king, and vice versa.

This argument is convenient because it presupposes that RHO ALWAYS performs a mental coin flip in this situation. By doing so, he optimizes the expected value of his falsecards. In the world of economics, this is known as rational choice theory. In the REAL world, however, humans are anything but rational. RHO's choice of playing the king or queen is determined not only by rational choice theory, but by behavioral factors as well.

Being of idle time, I found 220 OKbridge tournament deals in which a defender held both missing honors behind an AJT or AKT9 tenace. A little analysis revealed two major factors that influenced which honor a defender played to the first finesse. These factors are explained below.

Factor 1. Who led the suit?

# True Cards # Falsecards
Declarer 88 101
Partner 23 8
Totals 111 109

We usually apply the principle of restricted choice when watching DECLARER take a second finesse. In this situation, we see RHO play the queen 47% of the time (88:101). And yes, that's pretty much a coin flip.

But when a DEFENDER first leads into a double tenace, his partner is likely to follow the time-honored defensive rule of playing the lower of two touching honors. This occurs at a rate of 74.2% (23:8). If you're wondering about sample size validity, the probability of seeing only 8 falsecards or fewer in 31 trials is 1.1%.

To give you an idea of what a difference 74.2% makes, consider this holding:



West leads a low card, East plays an honor, and you win the king. You take the ace and cross back to your hand in another suit. Now you lead toward the Q9, and West follows with a low card. What are your a priori odds for finessing?

Traditional restricted choice would suggest that it's about 64% (6.5:3.6) to finesse, because East was 50:50 to play either honor from J-10-x:

East Probability
J from J-10-x 7.2% * 50.0% = 3.6%
10 from J-10-x 7.2% * 50.0% = 3.6%
Jack from J-x 13% * 50.0% = 6.5%
10 from J-x 13% * 50.0% = 6.5%

If we consider the probability that East is only 25.8% likely to falsecard his partner, though, the probabilities change thusly:

East Probability
Jack from J-10-x 7.2% * 25.8% = 1.9%
10 from J-10-x 7.2% * 74.2% = 5.3%
Jack from J-x 13.0% * 50.0% = 6.5%
10 from 10-x 13.0% * 50.0% = 6.5%

If East played the 10 on the first round, the likelihood of a successful second finesse is only 55.1%, or 6.5 / (6.5+5.3). Did East play the jack? If so, the chance of a successful second finesse zooms up to 77.4%, or 6.5 / (6.5+1.9).

You can now weigh your chances to gain an extra trick in another suit accordingly, such as one that requires a 68% 3-2 split.

Factor 2. Is the Defender Holding a Doubleton?

# True Cards # Falsecards
Doubleton 21 49
Tripleton or greater 90 60
Totals 111 109

The second major behavioral factor that influences a defender's decision to falsecard is suit length. You can see in the table above that defenders are highly likely to falsecard with doubleton honors. 49:21 translates into a 70% falsecard rate. When a defender holds a tripleton such as KQx, though, the falsecard rate drops to 40%. It's as if defenders "save" their falsecards for the times when they are in genuine danger of losing a trick.

The implication here is that the more cards that you hold in a suit combination, the more likely the probability of a KQ or QJ doubleton, and the more likely a falsecard will occur. Consider this classic suit combination:



You cash the ace and East plays an honor. Now you lead toward dummy. What are the odds for finessing?

East Probability
Jack from J 6.22%
Queen from Q 6.22%
Jack from QJ 6.78% * 30% = 2.03%
Queen from QJ 6.78% * 70% = 4.75%

If East played the jack, it's a 75.4% bet (6.22/(6.22+2.03)) that you should finesse. East will falsecard with the queen 70% of the time, NOT 50% as rational choice theory would suggest. If East played the queen under your ace, a second finesse is only 56.7% (6.22/(6.22+4.75)).

You can see that the odds in real life are clearly not the same as those in theory. As declarer, you can take advantage of these discrepancies to estimate more precise odds of success. And on defense - consider reviewing your own falsecarding habits, and mixing up your falsecards more frequently.

Note: I am NOT suggesting that these are the only factors that influence a double finesse. The defense's opening lead methods, tendencies to be "tricky" and other factors certainly apply.