A Question of Odds

First, how do you play this hand in 6 ♥ after the lead of ♠ K? Read ahead only after deciding on your line of play.

♠ 10 6 4
♥ A 6
♦ K Q J 10 3
♣ A 6 4

♠ A 8 7
♥ K J 10 7 5 3
♦ A 6
♣ K 5

Well, the correct line is to win the first trick in hand, then cash the two top hearts. If the queen hasn't appeared (but hearts are 3-2), you hope that the defender with long trumps also holds at least three diamonds, because then you can get rid of the losing spades from your hand.

But do you know why this is better than finessing the ♥ J after cashing ♥ A? Do you analyse these situations correctly?

Whenever you have an 8-card trump fit including A, K, and J, and the jack is finessable, give a thought to cashing honours from the top. Given that trumps are 3–2, the probability that the queen is doubleton to that that it's trebleton is 2:3. That means a difference of just over thirteen per cent (a fifth of 68 %, the probability that hearts will be 3–2). The finesse will win only half the time. So if you can “recover&rdquo the 7 % through a second chance not available when you finesse, you must reject the finesse.

How do we do here? The second leg of your plan comes into picture when the queen is trebleton. How often will it accompany three or more diamonds? Think of it this way: the long-trump hand can have 0, 1, 2, 3, 4, 5, or 6 diamonds, and the probabilities of these should be roughly symmetrical about the half mark, 3.5. Since 3+ diamonds is favourable to you, the chances of you succeeding in your second leg are over 50 %. More than half of what? Well, the three-fifths chunk of 68 % when the queen is trebleton, i.e. more than half of around forty per cent. (It turns out that this “backup” plan actually adds about 25.4 % to you chance of success, while it costs only about 7 % to have the backup available. So the correct line is markedly better.

It's surprising how well you can do with simple approximations if you remember a few basic odds. Sometime in the future, I'll list some common odds that I think it is useful to remember