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The Principle of Restricted Choice's really very simple...

The Principle of Restricted Choice has recently come under fire in the letters section of English Bridge, and I find it bizarre that some players don't understand it. Put simply, it says that any action taken in preference to a plausible alternative makes it more likely that the alternative was impossible. If, say, a defender plays one card rather than an equivalent one (and is not trying to signal), then it is about 2–1 that his partner had the equivalent card and he had no choice.

A standard example in bridge is the following suit combination, with West declarer:

Kxxx [W E] AT9xx

If West cashes the king and South plays the jack, West might reason that South was slightly more likely to have been dealt QJ doubleton than J singleton, and now play the ace to catch the queen. The PRC says "since South chose to play the jack, he probably had no option to play the queen", so West should finesse. Let us analyse the situation.

Il y a trois possibilités: 'oui', 'non', autre chose...

There are five ways that South could play an honour:

  1. play the singleton jack
  2. play the singleton queen
  3. play jack from QJ doubleton
  4. play queen from QJ doubleton
  5. play one from some other holding.
The probability¹ of South being dealt QJ doubleton is
22C11 / 26C13 = 6.78%,
and the probability¹ of his being dealt J singleton is
22C12 / 26C13 = 6.22%.

We know that the probability of 1. is 6.22%, as is the probability of 2. The total probability of 3. and 4. is 6.78%. We suppose he could plausibly choose either—let's say he wouldn't pick the same one more than 90% of the time. 5. does not concern us, as in this case either finessing or cashing the ace will have the same effect.

If West did not know which card South played, but merely that it was an honour, she would finesse—South was more likely to be dealt honour singleton than QJ doubleton (12.43% compared to 6.78%). If, however, she knew that South had played the jack, she should still finesse, since she is not comparing the probabilities of J singleton and QJ doubleton (1. against 3.+4.), but the probabilities of 1. and 3.—6.22% against roughly 3.39%.

After all, if you would play the same way whether South plays the queen or jack, why would you play one way if you knew which card it was, and the other if you didn't? Queen-jack colour-blindness certainly can't help you play the hand, although it may help you understand PRC.

¹ assuming that all we know is which cards EW hold