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Bridge is full of numbers and some are quite complex. The principle of restricted choice is hard to grasp, but it is easy to put into practice.

 A K 10 9 8 7 6 5 4

You lead the ace of spades and East drops the queen, what do you do next?

Put another way: did East start with a bare queen or queen-jack doubleton? The important layouts are:

 West East a) J 3 2 Q b) 3 2 Q J

A quick mathematical calculation may suggest that b) is a fraction more likely. However, this is where the principle of restricted choice comes in. What it states is that half the time East holds Q-J doubleton he would play the jack first not the queen, so this halves the chances that he started with Q-J. Look at this another way: if East has the Q-J, let him play them to the first two spades on four different occasions:

 (i) J then Q (iii) J then Q (ii) Q then J (iv) Q then J

He chooses randomly whether to play the jack or queen first. The principle of restricted choice takes into account that options (i) and (Hi) are impossible (as East followed with the queen first). So the chance that he started with Q-J doubleton is only half what it was and you should take the finesse. Knowing why is not necessary, but being able to follow the principle will enhance your declarer play:

When two key touching honours are missing and a defender drops one, it is more likely to be a forced play from a bare honour than a chosen play from two honours doubleton. So you finesse his partner for the second honour.

 A K 10 9 8 J 3 2 Q 7 6 5 4

So, having played the ace of spades and seen East’s queen, you come to hand to lead a spade to North’s ten. Here is a full deal:

 K Q 3 10 9 5 4 8 4 3 K 7 5 J 8 7 6 J Q J 9 5 J 10 9 8 10 9 5 Q 3 2 A K 10 2 Q 6 3 A 4 2 A K 8 7 6 7 6 A 4 2

Contract: 4

To justify the bidding you need to avoid a trump loser. The defence play three rounds of diamonds and you ruff the third round, all following. You cash the A, on which West plays the jack. How do you continue?

Using the principle of restricted choice, you play West for having started with a singleton honour. Go to dummy with a spade and lead the heart ten, running it when East plays low; your king drops the queen on the next round and you have ten tricks.
Here is another example:

 A Q 4 7 6 4 5 4 3 9 8 6 5 10 3 2 10 5 K Q J 9 8 K J 10 J 7 6 5 J 8 3 2 10 7 2 Q 7 K 9 8 A K Q 9 A 6 A 4 3 2

Contract: 3NT