Probability of Games & Slams

[Larry Lowell]

I don't know where you got the idea that 70% of hands are partscores. The
statistics I've heard are 50% partscore, 50% game, 10% slam (small or
grand). Take a look at the contracts that you play in on an average
session. I think you'll find the 50% partscores is a *high* percentage. I
certainly don't play that many contracts in partscore.
David

I thing these figures are much too high.

I did a quick spread sheet on power hands (no distributional
considerations) to calculate these percentages for hands of 26 or more
HCP. (I know 25 HCP is the standard for experts, but hey let's be
real.) Assuming that you should bid all 26 total hcp partnership hands
to game:

Game frequencies are about 20% (24% if using 25 hcp). Distributional
considerations would raise this, but that is very difficult to
calculate. Assuming a singleton is useful 25% of the time raises the
frequency to 23.5%. Match Point Bidding needs to emphasize part score
bidding as well as games.

For slams on power alone, the frequency is 2%. (33+ HCP - yes I know
33 isn't enough for balanced hands without 5-card suits.) Assuming a
useful singleton 25% of the time raises the frequency to 3.1%. Slams
are less important at Match Point Pairs than at IMP Teams. Just don't
miss the obvious ones.

My reference was Traub & Desai, 1990: The Role of Combinations in
Contract Bridge vis-a-vis Point Count Distribution or Foundation of
Mathematical Analysis of Contract Bridge. I have Borel's book also,
but it doesn't deal with hcp probabilities (on a quick search).

How about some SIMULATIONS to give a better answer that includes
distribution?

Larry Lowell
Knoxville, TN, USA

[Larry Lowell]

Post by Larry Lowell
I did a quick spread sheet on power hands (no distributional
considerations) to calculate these percentages for hands of 26 or more
HCP. (I know 25 HCP is the standard for experts, but hey let's be
real.) Assuming that you should bid all 26 total hcp partnership hands
Game frequencies are about 20% (24% if using 25 hcp). Distributional
considerations would raise this, but that is very difficult to
calculate. Assuming a singleton is useful 25% of the time raises the
frequency to 23.5%. Match Point Bidding needs to emphasize part score
bidding as well as games.
For slams on power alone, the frequency is 2%. (33+ HCP - yes I know
33 isn't enough for balanced hands without 5-card suits.) Assuming a
useful singleton 25% of the time raises the frequency to 3.1%. Slams
are less important at Match Point Pairs than at IMP Teams. Just don't
miss the obvious ones.
My reference was Traub & Desai, 1990: The Role of Combinations in
Contract Bridge vis-a-vis Point Count Distribution or Foundation of
Mathematical Analysis of Contract Bridge. I have Borel's book also,
but it doesn't deal with hcp probabilities (on a quick search).
How about some SIMULATIONS to give a better answer that includes
distribution?
Larry Lowell
Knoxville, TN, USA

Hmmm, I found my Frost book: bridge odds complete, probabilities in
contract bridge, 2nd edition, 1971.

The probability of game on power only, 26+ hcp is only 12.7% (17.5% if
25+ hcp). With a useful singleton 25% of the time, 22+ hcp the
probability is 18.9% (24.6% if 21+ hcp).

The probability of slam on power alone, 33+ hcp is 0.35%. With a
useful singleton 25% of the time it is 1.2%.

These figures ignore double fits and fits greater than 8-cards where
less hcp are needed for slam.

If partner opens the bidding, the probability for game increases
significantly.

Thus, Match Point Pairs Bidding Systems should be designed not only
for game investigations, but also for partial contracts since these
seem to be more frequent, 50% probably. Slam, on the other hand, are
less important for pairs (much more important for IMPs).

Larry

[David Stevenson]

Post by Larry Lowell

I don't know where you got the idea that 70% of hands are partscores. The
statistics I've heard are 50% partscore, 50% game, 10% slam (small or
grand). Take a look at the contracts that you play in on an average
session. I think you'll find the 50% partscores is a *high* percentage. I
certainly don't play that many contracts in partscore.
David

I thing these figures are much too high.

So do I . 50+50+10 seems too much of a percentage .... :)

Post by Larry Lowell
I did a quick spread sheet on power hands (no distributional
considerations) to calculate these percentages for hands of 26 or more
HCP. (I know 25 HCP is the standard for experts, but hey let's be
real.) Assuming that you should bid all 26 total hcp partnership hands

Let's not forget singletons and fits. I think the average hand bid to
game whether it makes or not is probably more like 24 HCP.

Of course, we have to be clear what the question is. Is it what
percentage of hands are bid to game+? - I would ask what percentage of
hands have a combined 24+ HCP. Or is it what percentage of hands can
make game? - now I think a simulation is the only answer, though I
expect it to be slightly higher.

[Reef Fish]

Post by David Stevenson

Post by Larry Lowell

I don't know where you got the idea that 70% of hands are partscores. The
statistics I've heard are 50% partscore, 50% game, 10% slam (small or
grand). Take a look at the contracts that you play in on an average
session. I think you'll find the 50% partscores is a *high* percentage. I
certainly don't play that many contracts in partscore.
David

I thing these figures are much too high.

So do I . 50+50+10 seems too much of a percentage .... :)

Hey, but that's average for this gorup.

Post by David Stevenson

Post by Larry Lowell
I did a quick spread sheet on power hands (no distributional
considerations) to calculate these percentages for hands of 26 or more
HCP. (I know 25 HCP is the standard for experts, but hey let's be
real.) Assuming that you should bid all 26 total hcp partnership hands

The problem with this approach is our old friend: Points/Schmoints.

You CAN ask the question based on an arbitrary number of combined
HCPs. But the percentage answer you get will be the answer to the
wrong question! Because, games and slams are not made on POINTS,
or SCHMOINTS.

Post by David Stevenson
Let's not forget singletons and fits. I think the average hand bid to
game whether it makes or not is probably more like 24 HCP.
Of course, we have to be clear what the question is.

NOW you're finally talking! What IS the proper question to ask?

Post by David Stevenson
Is it what percentage of hands are bid to game+?

That would be one question.

Bid to game by WHOM? An expert partnership? All other bidders?

Post by David Stevenson
- I would ask what percentage of
hands have a combined 24+ HCP.

That would be an entirely different question. See Schmoints.

Post by David Stevenson
Or is it what percentage of hands can make game?

That's still another different question! What do you mean by
"can make games"?

1. Double dummy?
2. Single dummy?
3. Against perfect defence?
4. Against normal but imperfect defence?
5. Can make a game in the most probable (modal <g>) game contract?
6. Can make game in any biddable/makable game contract?
7. etc.

- now I think a simulation is the only answer, though I

Post by David Stevenson
expect it to be slightly higher.

After you clearly pointed out that the question whould be clear,
you have left yourself a completely unlcear quesiton.

Which of (1) - (7) did you mean, and how do you propose to
simulate the percentage result for any one of them?


To keep it simple and realistic, the only reasonable way is to do
it empirically on a large number of hands in a tournament setting
of a particular level. Do a CENSUS of all the results of all the
hands played. It would be a tedious and boring job, but you would
get close to the realistic "percent" result you sought.

Simulation is NOT the correct approach to this ill-defined and
ill-formulated problem.

-- Bob.

[Larry Lowell]

Post by David Stevenson

Post by Larry Lowell

I don't know where you got the idea that 70% of hands are partscores. The
statistics I've heard are 50% partscore, 50% game, 10% slam (small or
grand). Take a look at the contracts that you play in on an average
session. I think you'll find the 50% partscores is a *high* percentage. I
certainly don't play that many contracts in partscore.
David

I thing these figures are much too high.

So do I . 50+50+10 seems too much of a percentage .... :)

Post by Larry Lowell
I did a quick spread sheet on power hands (no distributional
considerations) to calculate these percentages for hands of 26 or more
HCP. (I know 25 HCP is the standard for experts, but hey let's be
real.) Assuming that you should bid all 26 total hcp partnership hands

Let's not forget singletons and fits. I think the average hand bid to
game whether it makes or not is probably more like 24 HCP.
Of course, we have to be clear what the question is. Is it what
percentage of hands are bid to game+? - I would ask what percentage of
hands have a combined 24+ HCP. Or is it what percentage of hands can
make game? - now I think a simulation is the only answer, though I
expect it to be slightly higher.
David

I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

He did double dummy analysis on 900,000 hands:

N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

Also with favorable vulnerability, sacrifice is good 17.6 %
with unfavorable vulnerabiity, sacrifice is good 1.7 %

Best contract (NT or suit) average trick is 8.44

Larry Lowell
Knoxville, TN, USA

[Frans Buijsen]

Post by Larry Lowell

Post by David Stevenson
Of course, we have to be clear what the question is. Is it what
percentage of hands are bid to game+? - I would ask what percentage of
hands have a combined 24+ HCP. Or is it what percentage of hands can
make game? - now I think a simulation is the only answer, though I
expect it to be slightly higher.
David

I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!
Also with favorable vulnerability, sacrifice is good 17.6 %
with unfavorable vulnerabiity, sacrifice is good 1.7 %
Best contract (NT or suit) average trick is 8.44

These figures look to me like he was looking only at the NS cards, not
at the EW cards. On the 18.4 % of hands where NS can't make anything,
it's probably because EW have really good cards.

That would indicate that hands where both NS and EW cannot make more
than a partscore is something like 49% (the square of 50.8 + 18.4 %).
That's just an approximation though, since these variables aren't
independent.

[Mark Brader]

Post by Frans Buijsen

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Frans Buijsen

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

These figures look to me like he was looking only at the NS cards, not
at the EW cards. On the 18.4 % of hands where NS can't make anything,
it's probably because EW have really good cards.

Makes sense. It would be interesting to see the above extended to a
matrix vs. the same 5 cases for E-W.

Which leads to the question, will the cell where neither side can make
any contract contain a 0, meaning that this never happens? With only
900,000 deals I would guess that that's true -- but what about
constructed hands? Is it possible to construct a hand where, playing
double dummy, *either* side can take 7 tricks regardless of trump suit
or notrump if it has the opening lead, and therefore nobody can make
any contract?

Just curious.

[DavJFlower]

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

Interesting, but the matrix would be more interesting - for example:

What percentage of hands can (no side/one side/both sides make a game) ?

Dave Flower

[Peterh]

Post by DavJFlower

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

What percentage of hands can (no side/one side/both sides make a game) ?
Dave Flower

Am I missing something ? IT is only fair that
anything that NS can do , so can EW
So game or better is (2*30.8%)61.6% So only 38.4% of hands don't make game
or better.
On the hands where NS are making partials , EW are frequently making Game or
better .


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[DavJFlower]

Post by Peterh
Am I missing something ? IT is only fair that
anything that NS can do , so can EW
So game or better is (2*30.8%)61.6% So only 38.4% of hands don't make game
or better.

Yes you are missing something.

On 30.8% of boards N/W can make game or better
On 30.8% of boards E/W can make game or better

Assume that on 5% of boards (I have just made up this figure, but it is clearly
greater than zero) both N/S and E/W can make game.

Then on 56.6% of boards either NJ/S, E/W or both can make game

Dave Flower

[Peterh]

Heehee Thanks . I guess 100+% cant be part scores :))

Post by DavJFlower

Post by Peterh
Am I missing something ? IT is only fair that
anything that NS can do , so can EW
So game or better is (2*30.8%)61.6% So only 38.4% of hands don't make game
or better.

Yes you are missing something.
On 30.8% of boards N/W can make game or better
On 30.8% of boards E/W can make game or better
Assume that on 5% of boards (I have just made up this figure, but it is clearly
greater than zero) both N/S and E/W can make game.
Then on 56.6% of boards either NJ/S, E/W or both can make game
Dave Flower

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[Larry Lowell]

Post by Mark Brader

Post by Frans Buijsen

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Frans Buijsen

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

These figures look to me like he was looking only at the NS cards, not
at the EW cards. On the 18.4 % of hands where NS can't make anything,
it's probably because EW have really good cards.

Makes sense. It would be interesting to see the above extended to a
matrix vs. the same 5 cases for E-W.
Which leads to the question, will the cell where neither side can make
any contract contain a 0, meaning that this never happens? With only
900,000 deals I would guess that that's true -- but what about
constructed hands? Is it possible to construct a hand where, playing
double dummy, *either* side can take 7 tricks regardless of trump suit
or notrump if it has the opening lead, and therefore nobody can make
any contract?
Just curious.

The EW figures are almost identical to the NS figures posted above,
varying only in the second decimal (if at all). Don't forget, this was
double dummy analysis. Thus, the 50% partial figures would include
some deals where both sides can make a partial (just like in real
bridge). Likewise, the figure for games, 24% must include a very
small percentage of hands where each side can make game.

The Main Point of this thread is that BIDDING SYSTEMS for Match Point
Pairs should be designed for finding good partial contracts since
these are more frequent than games, and furthermore that partial
contracts are more important than slams and you can be a winner at
pairs even with mediocre slam bidding tools.

Larry Lowell

[Mark Brader]

Post by Larry Lowell

Post by Mark Brader

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

It would be interesting to see the above extended to a
matrix vs. the same 5 cases for E-W.

The EW figures are almost identical to the NS figures posted above,
varying only in the second decimal (if at all).

Well, of course.

Post by Larry Lowell
Thus, the 50% partial figures would include some deals where both
sides can make a partial (just like in real bridge). ..

The question is, *how many* are of that kind, and similarly for the
other 24 possible combinations.

[John Montgomery]

Post by Mark Brader

Post by Frans Buijsen

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

...

Post by Frans Buijsen

Post by Larry Lowell
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

These figures look to me like he was looking only at the NS cards, not
at the EW cards. On the 18.4 % of hands where NS can't make anything,
it's probably because EW have really good cards.

Makes sense. It would be interesting to see the above extended to a
matrix vs. the same 5 cases for E-W.
Which leads to the question, will the cell where neither side can make
any contract contain a 0, meaning that this never happens? With only
900,000 deals I would guess that that's true -- but what about
constructed hands? Is it possible to construct a hand where, playing
double dummy, *either* side can take 7 tricks regardless of trump suit
or notrump if it has the opening lead, and therefore nobody can make
any contract?
Just curious.
--
Mark Brader | "If the standard says that [things] depend on the
Toronto | phase of the moon, the programmer should be prepared
My text in this article is in the public domain.

It is not hard to construct deals where no one can make any contract. In
fact, one of the best-known constructions results in a deal where any
contract played
by any of the four players goes down two (if played at the one level). Let
each player have some 4441 distribution, with AKQJ in one of the suits.
Also let each player's AKQJ suit face his partner's singleton. Assign a
one-level contract to any of the four players. The opponents can always
cash the first eight tricks, regardless of whether the contract is 1C, 1D,
1H, 1S, or 1NT.

John Montgomery

[Mark Brader]

Post by John Montgomery
It is not hard to construct deals where no one can make any contract.
... Let each player have some 4441 distribution, with AKQJ in one of
the suits. Also let each player's AKQJ suit face his partner's singleton.

Ah, right. I think I'd actually seen that before. Thanks for posting it.

[Chris Ryall]

Larry Lowell wrote on "Probability of Games & Slams"

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm

.. what a nice site it is too! If our embedded mathematicians might
confirm these stats are reasonable I am minded to link to it as one of
our better Bridge Internet resource.

[Steve Willner]

[very old thread]

Post by Larry Lowell
I found an answer on Peter Cheung's Bridge Website
http://crystalwebsite.tripod.com/what_is_new.htm
N-S make a partial 50.8 %
N-S make a game 24.0 %
N-S make small slam 5.3 %
N-S make grand slam 1.5 %
N-S cannot make any contract 18.4 % ***** New information!

Matt Ginsberg published similar results in BW but considered contracts
for both sides. I can't find it now, but it was probably about 1997 and
probably as a letter or "Bits and Pieces" item.