Principle of Restricted Choice - Better Calculation of Odds

Better Calculation of Odds

This is an attempt at a more accurate calculation of the odds as explained in the previous section.

A priori, four outstanding cards "split" as shown in the first two columns of the table. For example, three cards are together and the fourth is alone, a "3-1 split" with probability 49.74%. To understand the "number of specific lies" refer to the preceding list of all lies.

The last column gives the a priori probability of any specific original holding such as 32 and KQ; that one is represented by row one covering the 2-2 split. The other lie featured in our example play of the spade suit, Q32 and K, is represented by row two covering the 3-1 split.

Thus the table shows that the a priori odds on these two specific lies were not even but slightly in favor of the former, about 6.78 to 6.22 for ♠KQ against ♠K.

What are the odds a posteriori, at the moment of truth in our example play of the spade suit? If East does with ♠KQ win the first trick uniformly at random with the king or the queen – and with ♠K win the first trick with the king, having no choice – the posterior odds are 3.39 to 6.22, a little more than 1:2, in percentage terms a little more than 35% for ♠KQ. To play the ace ♠A from North on the second round should win about 35% while to finesse again with the ten ♠10 wins about 65%.

The principle of restricted choice is general but this specific probability calculation does suppose East would win with the king from ♠KQ precisely half the time (which is best). If East would win with the king from ♠KQ more or less than half the time, then South wins more or less than 35% by playing the ace. Indeed, if East would win with the king 92% of the time (=6.22/6.78), then South wins 50% by playing the ace and 50% by repeating the finesse. If that is true, however, South wins almost 100% by repeating the finesse after East wins with the queen – for the queen from that East player almost denies the king.