[ceeb]

kgr, on Sep 30 2010, 02:02 AM, said:

... I wonder how far off both methods are:

Dealer: ????? Vul: ???? Scoring: Unknown

♠ KJT9 ♥ KQJx ♦ xx ♣ xxx ♠ Ax ♥ xxxxx ♦ AKQ ♣ AKQ

♠ is trumps.
LHO leads a small ♥, RHO return a ♥ and LHO ruffs.
LHO plays a random ♦ and RHO follows that suit.
You cash ♠A and lead a second round. All follow with small ♠'s.
 
Probability that LHO has ♠Q according to vacant space theory:
13-1-3:13-6-1=9:6=0.6
 
Probability if WRONGLY taking the ♦ play into account:
13-1-3-1:13-6-1-1=8:5=0.615385

==> Is anyone able to calculate the real probabilities?

Interesting case. Unlike Fred's original hand, we have not seen the entire spade-spot suit (the pseudo- or sub-suit consisting of small spades) so cannot accurately compute via vacant spaces. But we've seen a fair fraction of it, so how inaccurate is ignoring it altogether? Plugging into my probability calculator:

First calculation: Using only that hearts are 1=3, the ♠Q is 6:5 to be on the left. That I think is the vacant spaces calculation.

Second calculation: Using also the observation that LHO has at least 2 small spades, the ♠Q is 7:6 to be on the left.

Third calculation: Taking into account as well that RHO has at least 1 small spade, the ♠Q is 33:28 to be on the left

As decimal probabilities the above are 0.545, 0.538, 0.541.

In principle the ♦ plays tell us something as well -- no ♦ void about. That increases the last probability to 0.5413, but of course making a computation like that while ignoring inferences from the bidding and play is entering the twilight zone.

From the above computations it seems that the vacant space rule is quite accurate.

Note that at the opposite extreme if you misapply vacant spaces by counting every card seen, then you would always, half-way through a trick, think it is k+1:k that the player yet to play has any given card.

Quote

If Playing ♣/♦ before the 2nd trump then:
- for Vacant Spaces this will always be 0.6
- for the WRONG calculation this will be:
1 minor: 0.615385
2 minors: 0.636364
3 minors: 0.666667
4 minors: 0.714286
5 minors: 0.8

Sorry, don't understand that part.

Quote

BTW: Does it make a difference for vacant space theory if opps always play their small cards from low to high?

It can make big difference. Perhaps you could adjust the vacant space theory to account for it.

[ceeb]

kgr, on Sep 30 2010, 02:02 AM, said:

... I wonder how far off both methods are:

Dealer: ????? Vul: ???? Scoring: Unknown

♠ KJT9 ♥ KQJx ♦ xx ♣ xxx ♠ Ax ♥ xxxxx ♦ AKQ ♣ AKQ

♠ is trumps.
LHO leads a small ♥, RHO return a ♥ and LHO ruffs.
LHO plays a random ♦ and RHO follows that suit.
You cash ♠A and lead a second round. All follow with small ♠'s.
 
Probability that LHO has ♠Q according to vacant space theory:
13-1-3:13-6-1=9:6=0.6
 
Probability if WRONGLY taking the ♦ play into account:
13-1-3-1:13-6-1-1=8:5=0.615385

==> Is anyone able to calculate the real probabilities?

Interesting case. Unlike Fred's original hand, we have not seen the entire spade-spot suit (the pseudo- or sub-suit consisting of small spades) so cannot accurately compute via vacant spaces. But we've seen a fair fraction of it, so how inaccurate is ignoring it altogether? Plugging into my probability calculator:

First calculation: Using only that hearts are 1=3, the ♠Q is 6:5 to be on the left. That I think is the vacant spaces calculation.

Second calculation: Using also the observation that LHO has at least 2 small spades, the ♠Q is 7:6 to be on the left.

Third calculation: Taking into account as well that RHO has at least 1 small spade, the ♠Q is 33:28 to be on the left

As decimal probabilities the above are 0.545, 0.538, 0.541.

In principle the ♦ plays tell us something as well -- no ♦ void about. That increases the last probability to 0.5413, but of course making a computation like that while ignoring inferences from the bidding and play is entering the twilight zone.

From the above computations it seems that the vacant space rule is quite accurate.

Note that at the opposite extreme if you misapply vacant spaces by counting every card seen, then you would often, half-way through a trick, think it is k+1:k that the player yet to play has any given card.

Quote

If Playing ♣/♦ before the 2nd trump then:
- for Vacant Spaces this will always be 0.6
- for the WRONG calculation this will be:
1 minor: 0.615385
2 minors: 0.636364
3 minors: 0.666667
4 minors: 0.714286
5 minors: 0.8

Sorry, don't understand that part.

Quote

BTW: Does it make a difference for vacant space theory if opps always play their small cards from low to high?

It can make big difference. Perhaps you could adjust the vacant space theory to account for it.

[bucky]

ceeb, on Sep 30 2010, 09:30 PM, said:

Interesting case. Unlike Fred's original hand, we have not seen the entire spade-spot suit (the pseudo- or sub-suit consisting of small spades) so cannot accurately compute via vacant spaces. But we've seen a fair fraction of it, so how inaccurate is ignoring it altogether? Plugging into my probability calculator:

First calculation: Using only that hearts are 1=3, the ♠Q is 6:5 to be on the left. That I think is the vacant spaces calculation.

Second calculation: Using also the observation that LHO has at least 2 small spades, the ♠Q is 7:6 to be on the left.

Third calculation: Taking into account as well that RHO has at least 1 small spade, the ♠Q is 33:28 to be on the left

As decimal probabilities the above are 0.545, 0.538, 0.541.

Please note that LHO has at least THREE spades: ruffed once and followed trump suit twice.

[ceeb]

Oh. Then 6/11, 43/83, 51/98, 5001/9608 or
0.545 0.518 0.5204 0.5205.

[rfp]

I agree with Fred's mathematical point. Essentially, the spade spots
become a pseudo suit since their play involved no choice or decision,
i.e., the missing queen would always be withheld. Therefore, at the
moment of decision vacant spaces are equal if RHO has exactly two
more known cards in another suit.

However, there is a restricted choice element (I didn't see mentioned)
relating to _how_ the ruff was obtained. If leader was simply leading his
partner's suit, there is nothing here. But if he _chose_ to lead a singleton,
a trump holding of xxx is more likely than Qxxx. That is, a singleton lead
is automatic with xxx, but hardly so with Qxxx.

Under these circumstances, it's certainly better to play for the drop in
the 50-50 situation. And it may be better to extend playing for the drop
even further.

--
Richard Pavlicek
Web site: www.rpbridge.net

[dake50]

Are you expanding on Roudi's rules?

How much side knowledge to sway alternate play? EG. weak2 by LHO swings a 1-4 safety play to "best"