What to lead when you have a Loner.

[Richardb02]

Our team is down 7-8.
I ordered Alone R1 S3, in Clubs, holding:

S2 leads:

And I must follow. Should I play:
Or ?

[Wes (aka the legend)]

You did not say what the upcard was, but that's ok becuz we don't really need that information to answer your question.

If the upcard was the you should trump high.

If the upcard was the or you should trump low.

Why should you trump high with a bower if the upcard was an AC? To answer that question first you have to figure how how often the dealer is void in spades. Well if the dealer knows the convention to always void himself in Next even if he has a singleton Next ace then we know he will be void in Next around 74% of the time.

The math behind that number can be found in this thread: viewtopic.php?f=8&t=303

So if you trump low in this instance, your loner sweep will get ruined around 74% of the time as that is approx how often the dealer will overtrump you. So trumping low gets you a loner sweep around 26% of the time. If you trump high, the probability you'll get a loner sweep = the probability the dealer only has the Ace of trump which is (2C0 x 15C5)/17C5 = 3003/6188 = 48.53%

48.53% is > than 26%, therefore trump high.

But what about when the dealer doesn't know the "void himself in Next convention"? Well the math gets complicated becuz we don't know which way the dealer will go if he can void himself in multiple suits, but the gap between 48.53% and 26% is wide enough to strongly suggest that even if the dealer didn't play perfectly you're still better off trumping high.

Now let's move on to those times the upcard is the 9C or the 10C. This one is easy to explain: always trump low becuz if the dealer has a guarded Ace and a void in spades, you're never getting a loner sweep no matter what you do. And if the dealer does have a spade of course you'd want to trump low since you will never get overtrumped. So always trump low with this holding when the upcard is the 9C or 10C.

[Wes (aka the legend)]

Now let's move on to those times the upcard is the 9C or the 10C. This one is easy to explain: always trump low becuz if the dealer has a guarded Ace and a void in spades, you're never getting a loner sweep no matter what you do. And if the dealer does have a spade of course you'd want to trump low since you will never get overtrumped. So always trump low with this holding when the upcard is the 9C or 10C.

Just so nobody's confused, I should've said:

Now let's move on to those times the upcard is the 9C or the 10C. This one is easy to explain: always trump low becuz if the dealer has A-X in trump and a void in spades, you're never getting a loner sweep no matter what you do. And if the dealer does have a spade of course you'd want to trump low since you will never get overtrumped. So always trump low with this holding when the upcard is the 9C or 10C.

[Richardb02]

Thanks Wes. It is much easier to understand your explanation than what went through my head while playing. Even later when reviewing the hand, the same stuff kept going through my head. Your explanation and the forum help immensely.

[Richardb02]

I came across a similar situation, but instead of R1 S3, it is
R2 S3 Up 7-5
Down
S1
S2
S3
S4
I order Alone.
S2 leads
Should I play or ?

[Wes (aka the legend)]

Gotta gamble and trump low in this spot.

If one trumps low how often will he get overtrumped by the dealer?

Answer: First figure out the dealer's range of possibilities. If we assume the dealer will always pick up with 3 trump in the first round or 2 trump + 2 or 3 aces then the dealer's total range consists of hands with either 1 diamond or zero diamonds minus those specific hand combos with 2 trump + 2 aces.

Hands with 1 diamond in it: 3C1 x 14C4 = 3,003

Hands with zero diamonds: 3C0 x 14C5 = 2,002

Minus the 1 diamond hands that would've been 2 trump + 2 or 3 aces in the first round:

(3C1 x 3C2 x 11C2) + (3C1 X 3C3 X 11C1) = 528

3,003 + 2,002 - 528 = 4,477

So 4,477 hand combinations represents the dealer's total possible range.

Now we can figure out how often our hero gets overtrumped if he trumps low.

First of all we acknowledge the obvious fact that any time the dealer has 1 diamond they can't overtrump (3,003 - 528 = 2,475 combos), so we only need to focus on the 2,002 hand combos the dealer has no diamonds, and out of those hands we simply isolate those zero diamond combos that contain neither the Ah or the Kh:

3C0 x 2C0 x 12C5 = 792 combos

So the number of hand combos the dealer will have that CANNOT overtrump our hero is 2,475 + 792 = 3,267.

Divide that number by the dealer's total range and the percentage you get will be the odds the hero will NOT get overtrumped on the Kd lead: 3,267/4,477 = 72.97%.

So our hero is getting overtrumped 100-72.97 = 27.03% of the time.

That is an acceptable percentage to risk being overtrumped when you factor in that trumping high to avoid getting overtrumped is no panacea for the hero. I'll spare people the calculations but when the hero trumps high the dealer will then have an effective guarded Ace or King in trump around 35% of the time (That number is slightly inaccurate becuz out of laziness I didn't factor 2 trump + 2/3 ace hands out of the dealers effective guarded range, but it's close enough).

So since the hero still has a decent chance of getting in trouble even if he trumps high, might as well gamble and trump low and hope for the best.