Wednesday, October 21, 2015

bridge splits re-visited

A couple years back, I wrote on the chances of various "splits" in bridge (and explained why this is something bridge players care about) in this post, which also explains the math behind the chances.

However, in that post, I failed to include the possibility of 7 trumps being out, because it is fairly rare. Due to some poor bidding on my part, I found myself playing 4 spades last night, and my partner and I only had 6 trumps between us.  Here are the chances of the different splits of 7 trumps that are out, between the other two players.

4-3 split: 62.2%
5-2 split: 30.5%
6-1 split: 6.8%
7-0 split: 0.5%

For completeness, here are splits with 6 and fewer (from the prior post).
For hands with 6 trumps out:
3-3 split : 35.5%
4-2 split: 48.4%
5-1 split: 14.5%
6-0 split:  1.5%

For hands with 5 trumps out, we get:
3-2 split: 67.8%
4-1 split: 28.3%
5-0 split: 3.9%

For hands with 4 trumps out:
2-2 split: 40.7%
3-1 split: 49.7%
4-0 split: 9.5%

For hands with 3 trumps out:
2-1 split: 78%
3-0 split: 22%

For hands with 2 trumps out:
1-1 split: 52%
2-0 split: 48%

It's worth mentioning that these probabilities are unconditional.  Since the bidding that precedes playing any given hand gives some information, it is typically true that some splits can be ruled out or downplayed.  For example. in the 4 spade hand I played last night, a 5-2 or (especially) worse split seemed unlikely, because there was no double from the other side, so I would've put the chances of a 403 split far higher than the unconditional 62%.