## 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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