Math problem

[Winstonm]

Do you ever have the feeling that the world is a tuxedo and you are a pair of brown shoes?

[starfruit]

Here's my 2 cents :

 

1)We start by considering only the position where you have to make an immediate decision whether to finesse or drop

 

2)West has 11 cards while East has 10 cards

 

3)We then consider all the possible holdings left which satisfies the original assumptions (T from QTx, QT, Tx)

 

Here are the possible holdings (Bold denote cards played) :

 

West East

T         Qxxxx

Tx       Qxx

QT      xxx

QTx    xx

 

4)Next calculate the probability of these holdings (now showing only cards left)

 

West East   %

--      Qx    90/420 = 21.43% ----(1)

x       Q     110/420 = 26.2% ----(2)

Q        x    110/420 = 26.2% ----(3)

Qx     --    110/420 = 26.2% ----(4) *Note that they add up to 100%*

 

5)Now consider the 2 lines on the trump suit alone

 

Line 1) --> Finesse the J followed by K, and doing your ♦ stuff if still possible

Line 2) --> Play K for the drop, followed by ♦ discards if the Q doesn't appear

 

 

Line 1) picks up (2) immediately, with extra chances for (1).

However, you have no chances against (3) + (4) assuming they will cash their ♠A (which shd be obvious).

 

Line 2) picks up (3) immediately, with extra chances for (2) + (4).

However, you have no chances against (1) provided defender is alert to ruff your

5th ♦ with a small trump

 

6)Actual calculation of the odds :

Line 1) --> 26.2% + chances that RHO has holding(1) + ♦xxxx (x%)

 

Line 2) --> 26.2% + chances that RHO has holding (2) +♦xxxx(z%) + LHO holding (4) + ♦xxxx (y%)

 

7)Calculating x% and y% and z%

 

x has a priori of holding (1) which is 21.43%

So now LHO has 10 Unknown cards (1trump 1♣ 1discard) compared to RHO's 8 Unknown cards (he has 4trumps and at least 1♣)

Actual calculation = 8.17% x 21.43% = 1.75%

 

Compare with :

 

y has a priori of holding (2) which is already 26.2%

So now LHO has 9 unknown cards (3trumps 1♣) compared to RHO's 10 unknown cards(2trumps 1♣)

Actual calculation = 10.84% X 26.2% = 2.84%

 

similarly, for z:

West has 10 unknown cards (2 trumps 1♣) while East has 9 unknown cards (3 trumps 1♣)

Actual calculation = 10.84% x 26.2% = 2.84%

 

Clearly, y + z has higher% (more cases!)

 

Summary :

Since the trump T play CAN be from QTx + Tx (which "normal" defender isn't capable of doing), odds goes towards playing for good breaks (3-2 in trumps, otherwise 3trumps with 4♦s). This is essentially due to the probability of QTx offside and Qxx onside being very close.

 

Final : Line 1) = 26.2% + 1.75% = 27.95%

Line 2) = 26.2% + 2.84% x 2 = 31.88%

 

P.S. perhaps you're better off relying on your natural instincts (or what others call "table presence") LOLLLLLLL ;)

 

I think the ability to be able to falsecard from QTx matters alot here, just as what others mentioned. If you think it's a hard play, the odds actually tilts towards Line 1) since the extra chances for Line 2) will be decreased, making the 2 lines even closer.

 

Hope I haven't made any calculation errors, will check when I'm back home :P

[cherdano]

Starfruit, I don't know where you get these "90/420" etc. numbers from, but they are clearly wrong.

[starfruit]

They're based on before declarer decides whether to play K/J,

where West has 11 unknown cards while East has 10 unknown cards.

 

There are 2 remaining ♥s outside, so total possible combinations for their position

= 21 x 20 which gives 420.

 

For holding (1), East holds the remaining 2 cards and there are 10 x 9 possible holdings where the 2 ♥s will fall into those 10 "holes".

 

This gives 90 current /420 total...

 

Hmm. . . perhaps there's something wrong with this reasoning? :huh:

[cherdano]

They're based on before declarer decides whether to play K/J,

where West has 11 unknown cards while East has 10 unknown cards.

 

There are 2 remaining ♥s outside, so total possible combinations for their position

= 21 x 20 which gives 420.

 

For holding (1), East holds the remaining 2 cards and there are 10 x 9 possible holdings where the 2 ♥s will fall into those 10 "holes".

 

This gives 90 current /420 total...

 

Hmm. . . perhaps there's something wrong with this reasoning? :huh:

I see, this makes sense. However, since you are counting only specific spot card holdings that are still possible after the spot cards that have fallen so far, you have to weigh each of them with the probability that defenders would play the spot cards actually seen, i.e. to include the restricted choice of spot cards.

 

Let's assume the missing spot cards are 234 and RHO has played 2 and 3. The probability of that happening from an original holding of 432 is 1/3, but from an original holding of Q32 it is 100%. This means that T4 on the left is 3 times as likely as QT.

 

However, as far as the relative probabilities you get the same answer by counting all possible original holdings, including holdings that are excluded by now due to which specific spot cards have fallen. I.e. there is one QT holding but three Tx holdings, so the latter is 3 times as likely.

 

You also make a mistake later on: when you count vacant spaces you should not exclude the club cards - unfortunately it is somewhat difficult to explain convincingly why. (There is an article on this on www.rpbridge.net somewhere, though.)

[starfruit]

Hmm. . . I think I got what you meant for the first part.

But for the second part I'll think about it later when i'm free :huh:

[han]

Hmm, LHO would play the 10 from Q10x and 10x 100% of the time? OK! :P

 

I had Q10 and did think for a moment about playing the queen (before decarer played the heart though, in my memory I played the 10 smoothly). Declarer made the percentage play and went down. After the hand Justin said something interesting but unfortunately I don't remember the exact words. Something like "most people are biased to combinations".

[kenrexford]

It is bad enough to lack info on table feel aspects, especially when the "odds" issue might be tight.

 

Normally, this is not particularly relevant.  However, my gut leans me on this one to table feel and inference over pure mathematics.

Why do you assume stuff. Maybe Jlall is planning on writing a computer program to play bridge. Maybe he felt his way through the table with both hands and felt nothing. You never know.

 

I find these kind of comments pretty condescending...

See -- that is the problem with bridge programs that play bridge. The extra element of the game is hard to program.

[starfruit]

Alright I've given the original probability thing a small thought :

 

Doesn't the fact that LHO will always play T if he has it (as given from the original post) makes it even chance whether, after playing the T, that he has either Tx or QT or QTx?

 

At that point of time there's only 1 x outstanding so the probability of an original holding of Tx vs QT should be equal.

 

It can't be 2 against 1 for Tx against QT since one of the Tx case is eliminated after RHO has played it.

Similarly, if we compare Tx against QTx or QT against QTx for LHO, at first it seems that it's less likely for LHO to have QTx since that requires 2 of the remaining cards to be placed with him, but i think the fact that LHO still has 1 more card than RHO counters it.

 

I dunno, but I still feel that the way I calculate the odds for the the remaining trump position is correct (no offense) since it takes into concern that some original holdings are already out. . .

 

However, for your other point I can't really catch it for the moment (busy with work).

Will try to look it up when I'm free. :P

[Trumpace]

It is bad enough to lack info on table feel aspects, especially when the "odds" issue might be tight.

 

Normally, this is not particularly relevant.  However, my gut leans me on this one to table feel and inference over pure mathematics.

Why do you assume stuff. Maybe Jlall is planning on writing a computer program to play bridge. Maybe he felt his way through the table with both hands and felt nothing. You never know.

 

I find these kind of comments pretty condescending...

See -- that is the problem with bridge programs that play bridge. The extra element of the game is hard to program.

Agreed, but that is not point.

 

The point is someone comes to the forum asking for a directed and specific question and people respond with "no info", "without that info i hate this question" etc.

 

In a way, though, you are doing the right thing in that you do point out how odds could change based on table feel/bidding, which is good to know for any bridge player. If this was a beginner/int forum, I wouldn't have made any comments.

[hrothgar]

For christ sake... It seems pretty clear that Justin was interested in analyzing this as a purely technical problem. I don't see why folks feel that its necessary to inject all sorts of superfluous comments about whether or not a player was hitching. It seems like a completely un-necessary distraction...

 

Even if you believe that your superior table feel will provide you with some additional information this still doesn't mean that you can ignore the purely technical aspects to problem. Your table feel might shift the odds one way or another, however, you need to understand the technical problem to really understand whether how the line of play has shifted.

[kenrexford]

OK. I'll explain my question.

 

It seems that the finesse gains over the drop when hearts are 3-2, Queen onside, and RHO lacks four diamonds (in which case the finesse gains nothing IMP-wise).

 

It also seems that the finesse gains when hearts are 4-1, queen onside, and the person with the Queen also has four diamonds.

 

The drop gains when the hearts are 2-3, with the Queen doubleton.

 

The "drop" also gains when the hearts are 2-3 or 3-2 and the third heart joins the fourth diamond.

 

In all other cases, it seems that either line works or fails equally.

 

IF dummy was able to bid or show diamonds, it seems that a stiff diamond lead might have been a decent option. This seems to reduce greatly the likelihood of diamonds being 1-4, which mitigates against the situations when diamonds are 4-1 (or 5-0). It also seems to mitigate against the 4-1 options, as leading a diamond from four might also seem reasonable, let alone from 5.

 

If RHO was able to double an artificial 5♦ for a lead, the 5-0 is probably eliminated as an option.

 

If the opponents are white and us red, wild shapes seem mitigated, such that I can reasonably eliminate 6610 and similar patterns.

 

The point seems reasonable. Assume a small number of possibilities, perhaps 11. Pure math out of context might make one line 6:5 favorite. However, perhaps some deduction eliminates four of the mathematical options, at a 3:1 ratio. This reduces the pool of possibilities such that the odds are now 3:4 the other way.

 

This is not a matter of watching for hitches. It is a matter of reducing the field of possible layouts by those that are improbable, or more precisely by altering the normal weight to be given.

 

I just declared a hand recently in hearts. LHO had shown a "three-suited hand" and was known to be short in diamonds (stiff). As 4441/5431 were possible, I knew that hearts split 4-1 or 3-2. This information eliminated, obviously, 5-0 splits (the major could not be 5-card), 1-4 splits, 2-3 splits, and 0-5 splits. Thus, rough math suggested that hearts were 2:1 odds to be 4-1.

 

Had I known early enough that LHO held five clubs, the average expected heart length reduces to 3.5, roughly.

 

All of this affected my line. I erred by playing for 4-1 too early. I had time to get the count on clubs safely and would have done well checking this out.

 

Similarly, some less obvious indicators exist on any given deal. The failure to bid, the failure to opt for one lead over another, etc. We all know this, of course.

 

This was not meant to be condescending. Rather, I simply wanted to analyze whether inference from these sources might change the odds expectancy from pure math without any inference. It might well not be relevantto the question. For example, someone might have told Justin that one play or the other was clearly supported by odds, and the odds without inference might also prove that person's analysis wrong. Something like that.

[Halo]

I think this is the worst post I have seen in this expert zone.

 

First the poster gets it wrong and doesn't say so.

 

Secondly we get all this meaningless twaddle about events at the table that we have no information for.

 

Get a grip guys.

[cherdano]

I dunno, but I still feel that the way I calculate the odds for the the remaining trump position is correct (no offense) since it takes into concern that some original holdings are already out. . .

Well, I already explained earlier why that is wrong (ignoring the restricted choice among spot cards for RHO), but I am not on a mission to convince you -- nor am I offended :)