In the fourth article on the shuffler, we showed that the most common number of trump to be dealt is 3 (~32%). In this article, we are interested in studying hands where a player had a chance to pick with 3 trump and none of them are the jack of diamonds. We will limit the analysis to jack of diamonds partner hands and not double on the bump.
When studying these hands, we would like to understand whether a hand meeting certain characteristics qualifies as a picking hand. We will consider a hand a picking hand if the expected outcome is better when a player picks than when a player passes. It might be that the expected outcome of picking is negative but the expected outcome of passing is even more negative. In this case, even though our expectation is not good if we pick, we should still do it because our expected outcome is worse if we pass.
Data
The data we will use for this analysis is found in the table below. In the table, “position” refers to the picking order. If the player had the choice to pick first, the position is 1. If they were on the end, the position is 5. The pass columns are not filled out for position 5 since those hands would either result in a leaster, a schwanzer, or a doubler. It is out of scope to analyze whether the player is better off passing on the end, but based on the average points for those hands when the player picks, picking seems to be the right play.
The high card is simply the most powerful card in the player’s hand. When the high card is the queen of clubs, the player might have the queen of spades, too. The high card makes a difference in terms of picking percentage and expected points. Other possible important factors include the quality of the player with the hand and the fail situation (points to bury, likelihood of freeing suits after bury). We will break the data down using that information at a later date.
For the picking measures, each hand is only counted once since only one player can pick for any hand. Two players might pass on the same hand with three trump, so the same hand might be included in two different rows in the table in the passing figures.
| Picking Position | high_card | Pick Count | Avg Pick Points | Pass Count | Avg Pass Points | Pick % |
|---|---|---|---|---|---|---|
| 1 | Queen Clubs | 20,469 | 1.9 | 6,603 | -0.4 | 75.6% |
| 1 | Queen Spades | 13,040 | 0.8 | 9,229 | -0.5 | 58.6% |
| 1 | Queen Hearts | 6,276 | 0.1 | 11,318 | -0.8 | 35.7% |
| 1 | Queen Diamonds | 2,208 | -0.6 | 12,042 | -1.0 | 15.5% |
| 1 | No queen | 605 | -1.1 | 33,089 | -1.3 | 1.8% |
| 2 | Queen Clubs | 17,047 | 2.0 | 5,667 | -0.5 | 75.1% |
| 2 | Queen Spades | 10,614 | 0.8 | 7,264 | -0.6 | 59.4% |
| 2 | Queen Hearts | 4,936 | 0.2 | 8,910 | -0.9 | 35.6% |
| 2 | Queen Diamonds | 1,613 | -0.6 | 9,024 | -1.2 | 15.2% |
| 2 | No queen | 313 | -1.5 | 23,880 | -1.5 | 1.3% |
| 3 | Queen Clubs | 14,579 | 2.3 | 3,892 | -0.5 | 78.9% |
| 3 | Queen Spades | 9,185 | 1.3 | 5,060 | -0.6 | 64.5% |
| 3 | Queen Hearts | 4,277 | 0.7 | 6,271 | -0.9 | 40.5% |
| 3 | Queen Diamonds | 1,426 | 0.1 | 6,240 | -1.2 | 18.6% |
| 3 | No queen | 269 | -0.3 | 16,293 | -1.6 | 1.6% |
| 4 | Queen Clubs | 12,543 | 2.9 | 2,205 | -0.4 | 85.0% |
| 4 | Queen Spades | 7,546 | 1.7 | 2,945 | -0.6 | 71.9% |
| 4 | Queen Hearts | 3,982 | 1.3 | 3,575 | -0.9 | 52.7% |
| 4 | Queen Diamonds | 1,454 | 0.6 | 3,657 | -1.1 | 28.4% |
| 4 | No queen | 500 | -0.2 | 9,478 | -1.5 | 5.0% |
| 5 | Queen Clubs | 11,389 | 3.5 | 100.0% | ||
| 5 | Queen Spades | 7,655 | 2.5 | 100.0% | ||
| 5 | Queen Hearts | 5,055 | 2.3 | 100.0% | ||
| 5 | Queen Diamonds | 3,388 | 1.7 | 100.0% | ||
| 5 | No queen | 4,651 | 1.5 | 100.0% |
Analysis
Even when the high card is not a queen, the passer is never better off than the picker. Given how much better it is to pick with three to the queen of clubs, it is surprising that the player passes 25% of the time.
We can see how strong the fifth position is in this table. Even when the high card is not a queen, the expected points are about 1.5. Queen of clubs high hands expect a whopping 3.5 points per hand.
The table helps us quantify the effect of one unit of power in the high card. Stepping down from a queen of clubs high to queen of spades high is much larger than stepping from the queen of spades to the queen of hearts. Interestingly, the picking percentage changes more for the lower queens than it does when going from the queen of clubs to queen of spades. Player seem to feel disproportionately more confident going in with the queen of spades than they do with the queen of hearts. In both cases, it looks like picking is the right play with either card as the high card.
Further Analysis
There are a few questions that would be interesting to study in a follow-up.
- Do individual players either pick or not pick on a given type of hand, or are players influenced by some other factors? Other factors might include the quality of the other players at the table or the result of the previous few picking hands they had.
- Are there hands within each row that are not picking hands due to the fail in their hand? Some might argue, “I didn’t have anything to bury.” Is that a good reason to pass?
- Are these figures influenced by a selection bias? If better players tend to pick on these hands and worse players tend to pass, it might be that the better result for pickers would not be there if worse players picked and better players passed. We should break these figures down by player quality in order to understand whether this is influencing our analysis. We will do that when we have more data.