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%.  

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Ultimate Frisbee: to Huck or not to Huck?

I play a lot of Ultimate Frisbee, a game akin to football in that there are end zones, but akin to soccer in that there is constant action until someone scores.  In Ultimate, you can only advance by throwing the disc (so-called because we generally do not generally use Wham-O branded discs, which are called Frisbees). An incomplete pass or a pass out of bounds is a turnover, as is a "stall," where the offense holds the disc without throwing for more than 10 seconds.

In other words, in order for the offense to score, you need to complete passes until someone catches the disc in the end zone.  The accepted method of doing this is to complete shorter, high-percentage passes.  On a non-windy day, it seems fairly simple for at least one of your six teammates to get open and thus you can march down the field.  Of course, one long pass, or "huck," can shortcut the process and give your team the quick score.  Much like football, the huck is not typically done except in desperation (game almost over due to time or thrower almost stalled).

However, I am not at all sure this logic makes sense.  Suppose you need six short passes to advance to a score.  If your team completes short passes with a probability of 90%, you will score about 53% of the time (90% to the sixth power gives the chances of completing six passes in a row).  In other words, as long as the chance of completing the huck is more than 53%, you would have a better chance of scoring with a huck.

Thus, the relative chances of scoring via the two methods depends on three things: 1) chance of completing a short pass, 2) chance of completing a huck, and 3) number of short passes needed for a score.  The graph below shows the threshold huck completion rate (the rate at which it makes more sense to huck) for different short pass completion rates and always assuming 6 short passes is enough for a score and one huck is enough for a score.

Of course, this simple analysis assumes 6 throws equals a score, and it also leaves out a number of other factors.  For example, an incomplete huck confers a field advantage to the hucking team because the opposing team has to begin from the point of in-completion (as long as it was in-bounds).  On the other hand, it may not take long for the opposing team to figure out the hucking strategy and play a zone style defense that will lower the hucking chances considerably.