The last hand at men's bridge club Tuesdya night was interesting. I picked up a hand with 22 high card points, and since I had balanced distribution I bid 2 no trump. My partner, a solid player named Rich Geiger, thought a bit and then bid 3 no trump. When he laid down his hand he laid down an Ace, and I said, "that's enough for your bid right there." Then I was surprised to see him lay down 5 more points!
I have been wondering whether he should have bid 5 no trump, the idea being that if I have 24 points (the range for my bid is 22-24), then we have the 33 points needed to bid a small slam. Faced with a 5 no trump response by my partner, I would bid 6 with 24 points, pass with 22, and with 23 evaluate it based on 9's and 10's, and on whether I had a solid 5-card suit which I could expect to run.
On the surface, it would seem that Rich's failure to bid 5 NT would cause us to miss a small slam a third of the time. But upon further analysis his bid makes sense.
First, having the 33 points does not guarantee the slam will be made. Tired players, who aren't that good to begin with, could easily mess up a slam, especially at the end of the session when fatigue is strongest. So, let's say it gets botched half the time, dropping the instances of losing out down from 1/3 to 1/6.
The adage of a bird in the hand being worth two in the bush applies. A vulnerable game gets you 500 points, which would be assured if you bid only 3 NT with 31+ points. The bonus for making a vulnerable small slam is 750 points. So, you are risking 500 sure points in an attempt to garner an extra 750 points. The math on this is that you would have to make the small slam 40% of the time to break even on taking the risk. (40% of 1,250 is 500.)
But here is the real kicker. Pondering this it occurred to me that a 22-point hand seems much more likely to get than a 24-point hand. If so, then it is not 1/3 each, but some other number. After some initial brick walls, I was able to find some odds on the internet. It turns out that there are 1334 million hands with 22 points, to 711M with 23, and only 355M with 24. So, given that a hand contains 22-24 points, it will have 22 points 55.6% of the time, 23 points 29.6% of the time, and 24 points only 14.8% of the time.
This lends further support for Rich's decision, as the odds of me having 24 points were not a third, but only about 15%. This is further lowered if you assume, as in our situation, that you have 9 points in your hand. In other words, given that A has a 9-point hand, the odds that his partner B would have 24 when he opens 2NT go down.
The conclusion: Rich probably should have bid 5NT, but it wasn't a blatant error.