Sunday, November 16, 2008

Law of Restricted Choice (LRC) - An application of statistics!

Your best friend (nerdy statistician type) asks you what you believe the odds are that any one Oregonian picked at random will give birth within the next year. Your surmise it's close to zero. Your friend now gives you new information: The person is female and pregnant.You drastically revise your estimate based on the new information.

Monty Hall tells you the prize is behind one of three doors. As requested, you pick one door at random. He now shows you one of the remaining two doors which does not contain the big prize and asks if you would like to change your mind and choose the one door which is left. You now have new information. Of the remaining two doors, Monty would not show you the door which held the big prize. Your odds at getting the right door (and the big prize)improve from 33%to 66% if you switch.

Your favorite pro is playing bridge at the Portland regional. As kibitzer you watch him/her play a contract which contains the following card combination AT3 opposite
K98534. After playing the A, your pro sees a Q or J fall from RHO. Your pro has new information. Both opponents could not play low.

The three scenes above are classic examples of the branch of mathematics called "Bayesian statistics". Simply defined, it is a mathematical science for revising the probability of events based on new information.

Let's concentrate solely on the bridge application. What thoughts are going through your pro's mind when he/she looks at this card combination, plays the ace, and an honor flops from an opponent. No doubt the pro is wondering whether he/she should finesse LHO for the missing honor or drop Q/J doubleton.

This is a direct application of the law of restricted choice, a bridge play based on a concept of Bayes theorem .

However, before the pro decides on which play to make he/she mentally goes through the qualifying checklist.

1) Does RHO randomly play the Q or the J from Q/J holdings. If so, the law applies and he/she should finesse. If not, the law does not help him/her locate the missing honor.
2) Are the missing cards of equal value? In this case "Yes". The law of restricted choice has no merit when the missing spots are Q873.

Having one spot card fall versus another, does not indicate the nature of the splits nor helps locate the Q in the suit. If both of the above conditions apply, the pro will finesse LHO for the missing honor. If both conditions do not apply, the pro will not be able to apply the law of restricted choice to help him/her determine the splits, nor locate the missing honor.

The question begs, is the above card combination the only card combination on which you can apply this Bayesian-based theory called Law of Restricted choice (i.e. nine card fits missing the Q/J) to help you locate missing cards? The answer is, "No". This theory applies in many other situations aswell. It applies with eight and seven card fits provided the cards you are trying to locate are of equal value and opponents are known to pitch randomly from those equal holdings.

Example 1: You have an eight card fit missing QJ. You see one of these equal-value cards fall on trick one. Since there is the possibility your opponent is offering a false card from QJx, you have to eliminate that possibility first -- i.e. based on the bidding or other factors which indicate the distribution of the hand is this particular opponent likely to hold a doubleton in the suit.

If he/she is, apply the other criteria as well and you are likely to make a better decision when playing the suit.

Example 2: You have a seven card fit holding KQx opposite A8xx. When playing the K/Q you watch the JT fall from RHO. Odds on that the finesse of the 9 spot through lho is the proper play. Once again those lovely spots are of equal value and you've got to give opponent RHO credit for randomly playing cards from JT9 to hide the true nature of the situation.

Hope this is of some use to you as you face these complicated card combinations as declarer player! Plenty of websites which addressthe issue should you like to explore the concepts above more thoroughly.

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