Thursday, February 16, 2006

The lone Ace

Euchre QOD: “Euchre is fun and his TV show is great”

Euchre Haiku:
The powerful ace
Alone, a great trick taker
Matched, mostly gets beat

Euchre Palaver
“Lone Ace” – A non-trump ace in your hand that has no other cards in that same suit.

Savvy Strategies
So you're in the first position, sitting with a lone Ace wondering whether you should start with that card or not. Let's look at the probability of it winning the trick.

First, a few assumptions used for figuring out the probabilities. This makes things easier.

1. If your opponents can win the trick, they will.
2. If your partner can win the trick, she will not.

When you have a lone Ace in your hand, there are 43,758 different hand combinations that your opponents can have. This is found by counting all the ways 18 cards can be divided up in groups of 10. 18 is for the number of cards the opponents could have (24 – 5 – 1 (for the up-card)).

Now it’s just a matter of finding out all the different cases when either one or both opponents don’t have the suit you’re leading.

So, if you have a lone Ace in a strong off-suit, there are a total of 5 other cards in that suit. The opponents could therefore have 0,1,2,3,4, or 5 of these cards.

If they have 0 or 1 of those cards (both are short-suited) then your Ace will not win. For simplicity we will disregard the times when they also don’t have trump to beat your Ace.

To figure out how many of the 43,758 hands they have 0 trump you just count 13 things taken 10 at a time. 13 is the number of cards they could have minus all the remaining cards in your Ace’s suit (18-5). The result is 286. That means 0.65% of the time neither of your opponents will have a card in the suit you led. A rare event indeed.

More common will be the times when 1 of the opponents is short suited and the other has 1 of the cards in your Ace’s suit. This will also result in a loss (unless the opponent also doesn’t have trump which we’ll ignore). This happens 3575 times or a total of 8% of the time.

The opponents will have 2 of the suit 12,870 times. But sometimes one opponent will have both of those cards and the other will be short suited and thus able to beat your Ace. This will happen 5720 times.

Similarly the opponents will have 3 of the suit 17,160 times and will be able to beat you 1560 times. They will have 4 of the suit 8580 times and will be able to beat you 130 times and will have 5 of the suit 1287 and be able to beat you 2 times.

Totaling up all the times they can beat you gives
286 + 3575 + 5720 + 1560 + 130 + 2 = 11,273 times.

Which means you will win 43758 – 11,273 = 32485 or 74.2% of the time. Whew!

Things are slightly different when you have the weak lone Ace (same color as trump) but I’ll save that for another day.

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