## Problem no. 43, 2012

### The problem

From time to time I make simulations with my deal program *Scania BridgeDealer*, but the result from one I made this spring surprised me initially.

You have surely heard somebody say that __the risk for bad splits go up when the opponents bid__. It is correct, because if one of the opponents have wild distribution, say 7-4-1-1, it is likely that she would bid something at some point. And when she has so wild distribution, none of your suits break 3-2 or 3-3.

I dealt North-South two balanced hands, where 3 notrump depended on 3-3 diamonds, and then I gave West a weak two-bid in hearts. If you know nothing about the opponents' hands, a 3-3 break is roughly 35.5%, so I expected that figure to drop. I ordered the program to generate 2000 deals to do statistics on. To my surprise, there were clearly more 3-3 breaks, not fewer. I did a new simulation, with 10 000 deals, but the result was the same.

Eventually, I realized that the result was as it should be. My assumpition before I let the program work had been wrong.

### Solution

In my simulation North-South had seven cards in all suits except hearts, where they had five. If the only restricition I had put on West's hand was "six hearts", it is true that the chance for a 3-3-split would go down, because if West has 6-5-1-1 or 6-4-2-1 none of North-South's long suits will be 3-3; and if he has 6-4-3-0 only one of North-South's seven card suits will be 3-3. But my restricition was "a weak two-bid", or, rather, a classic weak two-bid: __a one-suiter with a good six-card suit__. Then, only two distributions are possible for West: 6-3-3-1 and 6-3-2-2.

If West is 6-3-3-1, he will have a singleton diamond one time out of three (since North-South have the same number of cards in spades, diamonds and clubs). If West is 6-3-2-2 he will have three diamonds one time out of three. Had 6-3-3-1 and 6-3-2-2 been equally likely, the chance for a 3-3-split would be 50%; but since 6-3-2-2 is more likely than 6-3-3-1 the chance for 3-3 in diamonds is somewhat below 50%.

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