1-1k 26% 0-2 24% E k 100% Ek 50% Eq 50% Distribution of honours East play 1st club

Principle of Restricted Choice (or Freedom of Choice??)

The Principle of Restricted Choice states that when a player follows suit or wins a trick with one of two equal cards the probability of the player having both cards is halved.

Let us look at the following hand which you are playing in 3NT.

West leads S10, east plays S7 and you win with SK. At trick 2 you play a club to C10 and East wins with an honour and returns S2 to your SA and you now have a choice of finesses for your ninth trick. The Heart finesse looks like a straightforward 50% shot. The two Club honours are initially split one of four ways (all roughly equal probability) but now we are down to two distributions (0-2 or 1-1 with East having the honour shown) and initially those options were 24% and 26%, so surely we are still close to an equal shot. To find out if that is true draw a tree diagram and list the events.

List of events
Event name Event description
1-1k West has Queen, East has King
1-1Q West has King, East has Queen
2-0 West has King and Queen
0-2 East has King and Queen
Ek East plays the King
Eq East plays the Queen
Es East plays a small card

The list of events tabulates all the events that can occur (simplified by ignoring the small probability events of either defender holding a void, or West holding a singleton honour).

The diagram sets out the position before we start playing a trick. We now have additional information from the first Club trick. Let us assume the honour played is the King.

The new diagram shows the only two sequences on the diagram that fit the facts that we now have and we have coloured these blue and magenta. The question is are the blue and magenta sequences of equal probability?

Let us play the hand 1,000,000 times. Follow the blue route. There is a 24% chance of setting off on the blue route, so 240,000 times we are on that route. On this route East, having both honours can play either card, so if East plays an honour at random 120,000 times the King appears and 120,000 times the Queen appears (the yellow route). But we know East has played a King, so 120,000 times we would expect to be on the blue route, the yellow route has diluted the blue route probability by one-half. There is a 26% chance of setting out on the magenta route i.e. 260,000 times we will be on the magenta route. But when we are on the magenta route East can only play the King, so all 260,000 times we end up with the play of King by East. We are left with our destination of event Ek arising from magenta route 260,0000 times and blue route 120,0000 so the odds are 260 : 120 (68.4%) that we are seeing magenta with the honours split 1-1 and the second finesse working.

It is therefore a better shot to play the second club finesse rather than take the heart finesse.

I raised the question in the title as to whether Principal of Restricted Choice is a suitable name. The hand above is from "Bridge odds for practical players" by Hugh Kelsey & Michael Glauert who argue that the term "restricted choice" is an unfortunate one. To talk about "restricted choice" is to put the emphasis in the wrong place. It is not the restricted choice of the player who holds one honour card but the free choice of the player who holds two that necessitates an adjustment of the odds. I leave you make up your own mind about that but it might help to change thinking about Restricted Choice to the more user friendly Freedom of Choice. If you associate Freedom Of Choice with extra branches in the tree diagram then it helps to emphasise the weakening of the second honour line.

Hand simulation

The hand is run 1,000,000 times. On each run we

The 1,000,000 runs should include approximately 747,703 valid distributions which in turn include 68.25% 1-1 honour splits (slightly different from the 68.4% quoted above as we are now fully taking into account the hands where West has a void or a singleton honour) . We can now declare the results

Distribution not valid
The honours divide 1-1
The honours divide 0-2
Total of valid hands
Club finesse works