1-1 or 2-0 Split?

Your opponents hold two cards in a suit, is the split 1-1 or 2-0? Without any clues from the bidding or previous play, one might assume that the chances of these two splits are each 50% (1-1 = 50%, 2-0 = 25%, 0-2 = 25%). This assumption turns out to be wrong. The 1-1 split turns out to be 52%. So if you are forced to guess, bet on the 1-1 split.

Does the above make sense? Let's say that the two cards are the 2 and the 3. The 2 can be in either unseen hand with 50% probability; similarly the 3 can be in either hand with 50% probability. If West holds the 3 (50% chance), then there is a 50% chance he/she holds the 2, for a 25% chance that West has both cards. Right?

Well, there are not just two unseen cards left. If 2-0 is a possibility, then there must be at least four unseen cards. These are 2,3,x,y, where x and y are cards from some other suit. What are the possible combinations of these four cards?

1. 2,3 and x,y
2. 2,x and 3,y
3. 2,y and 3,x
4. 3,x and 2,y
5. 3,y and 2,x
6. x,y and 2,3

Clearly 2,3 and 3,2 are the same hand. So, within one player's hand, the order does not matter. These then are the only six combinations. The probability of a 1-1 split is 2/3 and 2-0 (or 0-2) is 1/3. This is not 52% and 48%, because this occurs near the end of the card play. If there are 26 unseen cards (at the beginning of the card play) and you need to know if two of them (two aces or two missing spades) split 1-1 or 2-0, then the probabilities are 52% and 48%, exactly. I could show that chart, of all of the various combinations of hands, but it is fairly large. If there are fewer than 26 unseen cards, then the probability of the 1-1 split increases somewhat.

Let's try to verify the 52%, without building the entire table. This is often done by analogy with marbles, but it is just as easy with cards. Our cards are the 2,3. Where do they go? Well, the 2 goes to West with 50% probability (13 ways, out of the 26 ways that this card can be dealt to EW). Now there are 25 cards left, and the 3 may go to West or East. But there are 12 ways that it can go to West, and 13 ways that it can go to East. It is slightly more likely that the 3 will go to East. The probability that the 3 goes to West is 12/25. So the probability that West gets both is 6/25 or 24%. The probability that East gets both is 24%, and the probability of a 1-1 split is 52%.

And so, we bet on the 1-1 split. And that is of major importance when calculating odds in many card combinations (suit combinations).