Tuesday, April 18, 2006

Euchre Statistics - More about the doubleton Ace

Euchre QOD: “Numbers are tyrannical”

Euchre Haiku:
A doubleton Ace
Will win more than it loses
At least if it’s green

Euchre Palaver
“Guarded” – This is a situation when you hold the second highest card plus another one in that suit. Most often encountered when you hold the Left Bower and another trump. Your bower is said to be guarded because if the Right Bower is led, you can play the low card and your Left Bower is then promoted to the highest ranking card.

Euchre Stats and Facts
While pondering the discussion in yesterday’s post, I was determined to figure out the actual probability that there would be a void if you held the Ace and King of a green suit.

So this is what I came up with.

If you have those 2 cards, it means you have 3 cards that are not in that suit and the up-card is not in that suit. So, there are 18 unknown cards. 4 of them are in your suit and 14 are not.

Since there are 4 of your suit and 15 cards in the hands of the other players, there has to be at least 1 card of your suit in play. But there will be a void if only 1 is in play. There will also be a void if 2 are in play.

The total number of ways that the 18 cards could be distributed between the 3 other players is a simple combination of 18 things taken 15 at a time, for a total of 816 possibilities. That’s not to say there are only 816 hands that the other players can have. That is just to say that there are only 816 ways that the unknown cards can be distributed in play or not in play.

In these 816 situations, there can be either 4,3,2, or 1 card from your suit in play. To figure out how many is a simple combination problem.

4 are in play 364 times (combination of 4 things taken 4 at a time * combination of 14 things taken 11 at a time)

3 are in play 364 times
2 are in play 84 times
1 is in play 4 times

In the case of 1 or 2 cards in play someone has to be void which will be the case about 11% of the time.

The case of 3 or 4 of the cards in play is slightly more complicated. The distribution of these cards can vary. If 3 cards are out, there are 3 unique ways the cards could be distributed. There could be 3 in one hand and none in the other two. There could be 2 in one hand, 1 in another and a void in the other. Finally, there could be 1 card in each hand. In all but the last case, there will be a void. These voids happen 87% of the time or in 317 of the 364 distributions. That represents 39% of the total number of distributions.

Similarly, if 4 cards are out we can figure out the number of times a void happens by looking at all the possible distributions. I skip the details and state that 47% of the time that 4 cards are out, there will be a void. This translates into 21% of the total distributions.

So what does all this tell us? If you have a doubleton Ace, you can expect that it could be trumped about 71% of the time (11+21+39). If your partner happens to be the one who is void then you have no worries. But odds are better (about 48%) that your opponents will be the ones with the void and it’s likely to get trumped.

Bottom line: If you have the Ace / King of a suit, lead the Ace. If you have Ace / King and another, lead the King.


Kevin said...

Nice math. Upon first glance, it seems right. I'm too busy to check it in detail (unfortunately tonight's exam is calculus, not probability), but it looks good.

It's painful seeing people use bad probability.

Perry said...

Ya know, I appreciate calculus much more now than I did when I was in college. Of course, I never use it for anything and I'm a scientist!

Harvey "the Rabbit" Lapp said...

Without even checking the math, wouldn't it be better to simply lead the King (from AK) so that your partner trumps it if possible?
I mean, your partner and your RHO might be the only two voids at the table, but your partner may assume that your Ace lead is a singleton and discard junk to let it walk. Leading the King may be more likely to insight your partner to ruff. Maybe I'm missing the point? I may be barking up the wrong tree.
Anyway, I must also mention that a guarded Ace (A,x,x) and King (K,x,x,x) are possible, and fun to have on defense.

Perry said...

You make a good point and one I pondered as I went through the math. The reason IMO that the Ace is the better lead (when you have 2 of a suit) is that the chances that 2 people are void are pretty slim (less than 10% of the time). So, most of the time if your partner is void she will be void alone and is trumping an otherwise winning trick. If you have 3 in a suit or even 4 well the odds of multiple voids is much higher to the point where it makes sense to trump in.

But I could see an argument for leading the King if you want your partner to trump the trick. It's just that in most instances, leading the Ace will win more tricks than leading the King.