## Problem no. 36, 2009

### The problem

When I was young, I sometimes heard this advice: "Don't bid a small slam, missing ace and queen of trumps". What I now wonder is: Was this good advice.

### Solution

Even if it wasn't stated explicitly, all solvers have assumed you have the playing strength for slam and that the important thing is to play the trump suit for one loser. If the advice is good or bad, depends on how many trumps you hold.

If we start from the top with eleven trumps, it is clear that we should bid a small slam, since we succed not only when trumps are 1-1 (52%) but also when the player to the left of the king has both the ace and the queen (24%). All in all, 76% chance of success. Note: if you don't have an entry to the hand opposite the trump king (so that you can't lead towards the strength) or if the king is singleton (trumps are distributed 10-1), the chance of one trump loser goes down to 52%. But even then, you gain more points in the long run by bidding slam than by staying in game.

With ten trumps, the first question is if we can handle ace-queen-third onside. If we can, we make a slam if the queen is onside (50%) but also if the player to the left of the king has the ace singleton (13%). The chance for making a slam then is 63%. If we miss the ten when our honors are divided between the hands, or if we miss the nine when the king is in one hand and jack-ten in the other, we go down against all 3-0-splits. Then, we can only handle four of the 2-1-splits, so the chance for success is 52%. Once more, bidding slam is with the odds.

When we have nine trumps, slam is at best 50%. Then, we can't handle ace singleton onside (which we could with ten trumps), so if we play for the best chance and finesse, we need the queen onside. But 50% is the right figure, if and only if, we also handle a possible 4-0-split. If we can't (either because our spot cards aren't strong enough; or if we can't afford to pull so many rounds of trumps; of if our trumps are divided 6-2, so that we only can take two trump finesses), the chance falls below 50%. To stay in game in that scenario is mathematically correct.

The reasoning for nine trumps applies to eight trumps as well. Then, it is even worse to bid slam, since already a 4-1-split can be disastrous (not to mention a 5-0-split).

A final thing against slam, when we miss the ace, the queen and at least one spot card, is that we may pick up trumps and still go down because of a ruff. That risk is always a real possibility when the opponents have a sure entry with the ace of trumps. The risk is small, since it requires a singleton (and that suit led) or a void in the hand which hasn't the trump ace, plus two or more trumps, but it isn't neglible. And that risk is what should get you to stay in game in what oherwise is a fifty percent slam.

Conclusion: If you miss ace-queen of trumps, it is correct to bid slam if you have ten or eleven combined trumps, but stay in game with nine trumps or fewer. If you follow that advice, you play with the odds.

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