# What are the odds?

One deal from the last session of the CNTC Final on the weekend provoked some heated discussion among the commentators about the relative odds of two lines of play. Since the deal in question was a grand slam, it was of no little importance to the declarer involved. This was the situation in the key suit:

West | East | ||||

♠ | ♠ | ||||

♥ | ♥ | ||||

♦ | ♦ | ||||

♣ | A K J | ♣ | 7 6 3 2 |

Declarer, playing in 7♥, needed 3 club tricks (obviously without giving up any!). Since he had lots of entries both ways and an available pitch from the West hand, he had two possibilities: a) cash the ace, then take a finesse b) cash the ace and king, then if the queen has not appeared, take his pitch and ruff a club. So the question is — with no other information, which is the better line? And for a bonus, is it significantly better, or is it close?

Most people I’ve posed this question to have opted to try the ruffing line — partly, I think, because no-one wants to make a grand on a finesse, and partly because there’s always the hope of some sort of squeeze if you ruff the club (although on this deal there wasn’t — clubs represented your only possible source of a thirteenth trick). Linda not only selected that line, but offered the opinion that it wasn’t close. Hence this blog. I mentioned on the BBO commentary that the odds were in fact pretty much the same — and everyone disagreed, including a number of spectators. One of them continued the discussion by email, eventually conceding gracefully the next day!

So here’s how it works. Assuming you have no other information (and I’ll get to that in a minute), the *a priori* odds stack up as follows:

a) cash the ace then take a finesse

You win any time South has the queen (50%) plus a little vigorish for a stiff queen in the North hand (one twelfth of 14.53% for all the 5-1 breaks, or 1.21%) — so 51.21 %.

b) cash the ace and king, then if the queen has not appeared, take the pitch and ruff a club.

You win with a stiff queen in either hand (one sixth of the 5-1 breaks, or 2.42%), a doubleton queen in either hand (one third of the 4-2 breaks, or 16.15%), and clubs 3-3 (35.53%). Add those up and you have 54.1%.

As I said above — pretty close. So is a 2.89% difference enough to hang your hat on when you’re playing a grand slam? Bob MacKinnon discusses this in his recent book, *Bridge, Probability and Information*: He says, ‘When the odds of two alternatives are nearly 50-50, in a practical sense it doesn’t matter which one is chosen, because the uncertainty is so great.’ You need to take into account everything that has happened on the deal, from the auction and opening lead onward, then take your best shot. It’s rare indeed that this kind of decision is purely a matter of *a priori* probabilities.

In the real-life deal, it certainly wasn’t, for two reasons. The more practical one was that while we discussing the right line of play in 7♥, the players actually bid to 7NT, where only one option was available — the finesse. Fortunately for declarer, the other reason was that North had preempted over West’s forcing club opening bid and was marked with six or seven diamonds. Thus by the time declarer had to play clubs, the finesse was much better than a 50-50 shot owing to the imbalance in Vacant Places between the North and South hands.

So the moral here is two-fold: if you run into this fairly common situation at the table, it’s pretty much a toss-up — and you should look for every other piece of information you can find to help you choose the line most likely to succeed.

On the actual hand the 4 small included the 10 …. which may influence the odds since dropping the Q now gives you the extra trick.

I was watching 7 hearts, suggested the ruffing line and was told it was ridiculous!!

Ray:

Why don’t you recreate the actual hands held by the declaring side.

Judy

West:

A2 AK87653 3 AKJ

East

108 Q942 AK2 10532

Having the 10 doesn’t actually make any difference to the odds here — you cash the AK before trying the pitch and ruff, so having the jack is enough. Once you take the pitch, you’re playing for 3-3 clubs, so in the end, the 10 doesn’t matter.

I think you forgot to add 1/2 of 6-0 breaks to the ruffout line (I make it 54.84 percent).

A more important issue that makes the ruffout line a standout (assuming the long heart hand declares) is the club-finesse odds are diminished by restricted choice; i.e., leader might have led a club without the queen but not with it.

This hand cannot be discussed in a vacuum. At the relevant table, the hand with C-AKJ dealt and opened a strong 1C, and the next hand bid 2D.

Also note that the opponents have two 9-card fits, one of them spades, yet spades were never bid.

Assuming, for the sake of argument, that the 2D overcaller would never hold 5S or 6C, and would always hold exactly 6 diamonds, the odds of his holding various club lengths are as follows:

5 – 2.17%

4 – 16.28%

3 – 43.42%

2 – 33.16%

1 – 4.97%

0 – never (he would have to hold 5S)

With this probability table for club breaks, playing a high club then taking the finesse succeeds 54.58%. Playing for the ruffout succeeds 61.09%.

When you factor in Richard’s point that, some of the time, N would have led a club from a low doubleton (singletons are irrelevant to the above calculation) the ruffout becomes even more attractive. In any case, a 6.5% edge is hardly to be sneezed at.

Note that my assumption that the overcaller would heave exactly 6 diamonds, and would not have as many as 5 spades, means that N has at most 10 pointed suit cards and S at least 8. This is why the likelihood of a 3-3 club break increases from the odds with two silent opponents.

Others making different assumptions would likely get different odds on various club breaks.