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Jeff Reubens Math Question

In his book, Expert Bridge Simplified, Jeff Reubens posed the following "Gedanken Experiment"

North:  2 / T98 / 9876 / 98765

South: --- / AQJ / AKQJT / AKQJT

West leads the Spade Ace. East shows out. Question What is the prob that East holds the heart King? Answ 13/14.

This is obviously correct as there are 14 outstanding cards and East has 13 of them.

Jeff then goes on to say that after West has played 12 spades, and East has discarded, what is the prob that East has the Heart King?

His answer is, Still 13/14. East Cards are 'shown' not known.  I have the highest regard for JR's math prowess but here is my problem.

In applying Vacant Places, JR in the same book, says that once the complete layout of a suit is known, we can treat the cards in that suit as 'known' not shown, and reduce the Vacant Places accordingly. So that after East has discarded 4 diamonds, can we not say that East now has 9 VP's down from 13? Or put it another way, after trick 1, West might have been dealt a diamond instead of the heart King. Now that we have eliminated that possibility, do not his chances of being dealt the heart King go up?

Moving on. East has now discarded his 3 clubs. So his VP's are now down to 6 from 13 by the same reasoning.

So are the odds of the heart King with West not increased now to 1/7 instead of 1 out of 14?

The heart suit is less clear cut. But again in JR's (and Kelsey's) book they state that when all but one of a suit have been played, we can count the played cards as 'known' for the purposes of VP calculations. Also the low (non King) hearts are what JR calls a 'subsuit' and can be treated like any other suit. By that reasoning at trick 12, East will have discarded all but 1 of the low hearts. So that further eliminates certain alternatives that might have been dealt to West. So East's VP's are now down to only 1, the same as Wests. Either East has one low heart left, or West does.

Since so many of the cards West might have been dealt have been eliminated, does that not increase the chance that he was dealt the heart King?

JR says absolutely not. We knew after trick 1, that East would discard 12 non heart King cards. So if the odds are now 50:50, then why were they not 50:50 after trick 1? that makes a lot of sense.

But I am having a hard time seeing how that answer fits in with the rest of the VP theory described in his book.

Anyone have an answer for this?

Greg Morse

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