[Fluffy]

I'd like a simulation towards this:

LHO is 18-19 Balanced, RHO has 6-7 HCP

LHO has exactly 2 spades, RHO has exactly 5 spades


You are missing spade queen, what is the percentage in each hand?


and if you are on to it, you can try missing ♠A, ♠K, ♠J to see the difference.

Thank you.

[Siegmund]

It'd sure be nice to have a rule of thumb for these, wouldnt it? Odd how humans handle these choices so well when they depend only on suit lengths but we lack a good way to cope with conflicting high-card and suit-length evidence. Maybe if I am bored this summer....

Anyway, a quick sim - 10,000 deals -> ~6k cases where their side holds a key card:

Ace:
Strong hand: 4366 (67.2±1.1%)
Long hand: 2131 (32.8±1.1%)
King:
Strong hand: 3015 (50.8±1.3%)
Long hand: 2924 (49.2±1.3%)
Queen:
Strong hand: 2062 (36.2±1.2%)
Long hand: 3629 (63.8±1.2%)
Jack:
Strong hand: 1529 (28.7±1.3%)
Long hand:3794 (71.3±1.3%)

Not that surprisingly, for the card worth the most HCP, the ratio isn't far from the 3:1 ratio of HCP-strengths, and for the card worth the least, it isn't far from the 2:5 ratio of suit-lengths ... but I'd have to stare at it a lot longer, with more other cases, to find a pattern.

Edited to add - this is the type of statistic that will change a LOT if AQxxx xx xx xxxx and xxxxx Ax Qx xxxx are not bid the same way - so it is going to be a bit hard to generalize to other auctions.

[Fluffy]

Thank's dude, my instinct was wrong, I though the Queen was the closer to 50-50 .

If you force LHO to have a clear lead such as ♦AK or ♣KQJ would it change much?

[Siegmund]

Very little difference. For 1000 hands I got a 64-36 split on the queen after forcing CKQJ.

My intuition was that all of them would be closer to 50% too (you would like to think you could take 3:1 points, 2:5 cards, and multiply to get 6:5 in favor of the strong hand as a good rule of thumb -- but that kind of shortcut only works when all the missing cards are of equal value, e.g., "East has 2 cover cards of the six possible ones I havent seen yet.")

[Cascade]

I got these numbers:

Missing ♠J 35.38% - 64.62%

Missing ♠Q 41.72% - 58.28%

Missing ♠K 52.04% - 47.96%

Missing ♠A 64.73% - 35.27%

Here is the code I used:

hcp(west)>=18 and
hcp(west)<=19 and
spades(west)==2 and
shape(west, any 4432 + any 5332) and

hcp(east)>=6 and
hcp(east)<=7 and
spades(east)==5 and

hcp(west,spades)+hcp(east,spades)==1 and

(hascard(west,JS) or hascard(east,JS))

[Siegmund]

This line looks like an error to me:

Quote

hcp(west,spades)+hcp(east,spades)==1 and

If that means what I think it does, it means you have forced E-W to hold the SJ AND to NOT hold the SQ,SK, or SA. It doesn't look to me like you should impose any restrictions on their honours except to reject the cases where they don't have the key card.

My code (for the Thomas Andrews dealer):

sdev na
sdev sa
sdev nk
sdev sk
sdev nq
sdev sq
sdev nj
sdev sj
source format/none

main {

reject unless {[balanced south]}
reject unless {[spades south] == 2}
set h [hcp south]
reject if {$h<18} {$h>19}
reject unless {[spades north] == 5}
set h [hcp north]
reject if {$h<6} {$h>7}
set w [whogets as]
if {$w == "south"} {sa add 1} 
if {$w == "north"} {na add 1} 
set w [whogets ks]
if {$w == "south"} {sk add 1} 
if {$w == "north"} {nk add 1} 
set w [whogets qs]
if {$w == "south"} {sq add 1} 
if {$w == "north"} {nq add 1} 
set w [whogets js]
if {$w == "south"} {sj add 1} 
if {$w == "north"} {nj add 1} 

accept
}

deal_finished {
set an [na count]
set as [sa count]
set kn [nk count]
set ks [sk count]
set qn [nq count]
set qs [sq count]
set jn [nj count]
set js [sj count]
puts "$an $as; $kn $ks; $qn $qs; $jn $js"
}

[Cascade]

That would have made my simulations run faster.

However I interpreted this:

"You are missing spade queen, what is the percentage in each hand?"

to mean you have the ace, king and jack. So it would be relevant to a play problem where you had AKJ and have the option of finesse or drop.

Maybe I am wrong. Fluffy can clarify his intention.

[Cascade]

Awaiting clarification of Fluffy's intent see above but I reran the simulations for the case where the strong hand has an AK or a KQJ in some side suit.

The numbers all dropped for the probability of the strong hand having the missing honour as follows:

Missing ♠J 33.94% - 66.06%

Missing ♠Q 40.45% - 59.55%

Missing ♠K 45.93% - 54.07%

Missing ♠A 54.56% - 45.44%

Here is the relevant additional code:

((hascard(west,AH) and hascard(west,KH)) or
(hascard(west,AD) and hascard(west,KD)) or
(hascard(west,AC) and hascard(west,KC)) or
(hascard(west,KH) and hascard(west,QH) and hascard(west,JH)) or
(hascard(west,KD) and hascard(west,QD) and hascard(west,JD)) or
(hascard(west,KC) and hascard(west,QC) and hascard(west,JC)) )

[Siegmund]

I can see where that's a possible interpretation. We do need to wait for clarification.

For the followup re natural leads, I got confused who LHO was, which didn't help. Heh. (I was forcing CKQJ to the left of the 18 pointer, not in the 18 point hand.)
If you're both forcing side honours in AND forcing spade honours out you are putting an awful lot of contraints on the 18 pointer hand, yes, and would expect a goodsized shift in the odds.