“It’s space, Number One, but not as we know it!” The First Officer winced: had the Captain at last understood the Theory of Empty Spaces, which states that the likelihood that a player holds a particular card is directly proportional to the number of unknown cards held? “You are probably thinking about the Theory of Empty Spaces, but I am going to tell you how to work out the changing odds of suit distributions as a hand progresses.” The First Officer’s ears pricked up. “Interesting,” he said.

“Yes,” continued the Captain. “This was explained to me by the Zian Commander after last night’s match in the Inter-Galactic League. Do you recall Board 34?” The Captain flicked to the hand on HoloGraph:

Board 34. N/S Game. Dealer East.

West

♠

53

♥

9842

♦

Q63

♣

KQ108

North

♠

A

♥

AKQ1075

♦

954

♣

952

East

♠

QJ10942

♥

J3

♦

KJ8

♣

J6

South

♠

K876

♥

6

♦

A1072

♣

A743

D

The First Officer remembered it well. The Zian East had opened a weak 2♠, which was passed round to him as North, where he had bid 3♥. The Captain bid 3NT, which was the final contract and West led the ♠5. At trick 2, the Captain had led a Diamond to the 10, won by West, who continued with Spades. Since holding up was pointless, the Captain had won, tabled the ♥6 and tranced before calling for the ♥10. Whereupon the roof had fallen in: East won, cashed four Spade tricks and switched to Clubs. The Captain ended with just four tricks for -500. When they scored up, it transpired that the Zian Commander had made ten tricks in the same contract for +630, producing a swing of 15 IGMPs to the Zians, who won the match by just 3 IGMPs. If only the Captain had let him play in 3♥.

“I took the opportunity to discuss the hand with the Zian Commander in the Officers’ Mess afterwards,” said the Captain. “She said that the bidding and the play to the first three tricks was the same, but that she had rejected the Heart finesse and simply cashed six Heart tricks and the two minor suit Aces to go with her two Spade tricks. I said I had played with the odds according to Theory of Empty Spaces: West presumably had 2 Spades to East’s 6, so West had 11 unknown cards that might be the ♥J, whereas East only had 7. So the odds were 11:7 on taking the Heart finesse. She said that my analysis was superficial, and the odds actually favoured playing East for ♥Jxx or ♥Jx. I asked her to prove it, and this is what she said:

‘After the defenders knocked out my second Spade stopper, the contract can not be made if either defender has five or more Hearts. If this had been the case, the best I could have done was to hope that West had the Jack and take the Heart finesse. But I could still only have cashed four Heart tricks and the two minor suit Aces for eight tricks and one down. So I needed Hearts to break 4=2 or 3=3. If East held three or four small Hearts, it made no matter what I did: I would always make six Heart tricks. Likewise, if East held ♥Jxxx, there was nothing I could do: I would be restricted to making only three Heart tricks. So the critical holdings were: ♥Jxxx with West, in which case I must finesse; and ♥Jxx or ♥Jx with East, when I must play for the Jack to drop.’

‘Now, I am going to let you into a secret, Captain. I can work out the comparative odds of the 4=2 and 3=3 Heart splits at the point that I needed to make the decision. This is how it is done. As you said, the Spade split is known, and West had 11 unknown cards to East’s 7. I needed to compare the 4=2 and 3=3 holdings, so, either way, West had to hold 3 of the defenders’ 6 Hearts and East must hold 2. This reduces their Empty Spaces for the 6th Heart to 8 and 5 respectively. Now comes the clever bit. West had 8 Empty Spaces, and if he were to have been dealt the final Heart, his suit length would be 4. The comparative odds that West did have those 4 Hearts are his Empty Spaces divided by his number of Hearts: that is 8/4 or 2. Turning the East: he had 5 Empty Spaces, and if had been dealt the final Heart, he would hold 3. So the comparative odds that East held 3 Hearts is 5/3. Thus it was slightly more likely that Hearts split 4=2 than 3=3 by 2 to 5/3. I am sure you learned cancelling and cross-multiplying in Elementary Space School, Captain, and prefer to express this ratio as 6:5.’

‘Next, I considered the likelihood of the three critical holdings. ♥Jxxx with West is two-thirds of all the possible 4=2 splits, so with comparative odds of 6, this had a likelihood of 4. So that was 4 for the finesse. ♥Jx with East is the other one-third of the 4=2 splits: therefore this had a likelihood of 2. I needed to add to this the likelihood of East holding ♥Jxx, which is one half of the 3=3 splits. With comparative odds of 5, this had a likelihood of 5/2 or 2½, making a total likelihood of 4½ for dropping the Jack offside. To summarise, the likelihoods of success were: 4 by finessing, but 4½ by playing Hearts from the top. As a Starship Captain, I am sure you appreciate that, with nothing else to go on, I simply played with the odds of 9:8 that the Jack would drop.’ ”

“I was dumbstruck,” the Captain continued, “and I can see by your face that you are, too.” The First Officer composed himself and replied, “It is an elegant method, Captain, and the arithmetic involved seems quite straightforward, but is it accurate?”

“Well,” said the Captain, “I left the Mess and went straight off to find Midshipman Math for confirmation. I asked him to calculate the probabilities of the Heart distributions once Spades are known to split 2=6, and in particular the ratio between the 4=2 and 3=3 splits. He thought for a moment and said that the probabilities are 37.33% and 31.11% and that the ratio between them is precisely - yes, precisely - 6:5. I then told him about the Zian Commander’s method, and asked if the whole *a priori* table of card distributions could be produced by means of her pair-wise comparisons. After a few seconds, he said that indeed it could, and, moreover, that the whole table could be expressed as ratios of whole numbers. As an example, he told me that the 3=2, 4=1 and 5=0 *a priori* distributions of 5 cards are in the ratio 156:65:9.”

“Fascinating.” said the First Officer. “Ensign ‘Flux Gun’ Kelly is in the brig for attempting the Alcatraz Coup against the Veegon Ambassador in last week’s Inter-planetary Pairs. I will give him the task of using the Zian Commander’s method to calculate all the ratios for the complete a priori table by hand.”

“Good idea, and could you demand another table for when one suit is known to split 6=2.” added the Captain.

After a short pause, the First Officer continued, “That Board 34 reminds me of one that I was reading about in the Historical Archives. It was back in the early 21st century, 2013 I think, in the United States trials on Planet Earth. The Heart position was very similar, and the unfortunate declarer took the finesse and ended up going 7 down in 7NT. At least we did better than that!”

Copyright © 2018 David Burch All rights Reserved

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