Restricted Choice Fallacies
#101
Posted 2016-July-18, 07:16
Bayes' Theorem is p(B|A) = p(A|B)p(B)/p(A).
For the problem set by Stephen, one way is to use:
A = LHO has xx, RHO has at least one honour
B = LHO has xx, RHO has KQ
So p(A) = 0.1922, p(B) = 0.0678, p(A|B) = 1 and p(B|A) ~= 0.35
We could of course also calculate the odds of the second honour being on the left instead by using:
C = LHO has Hxx, RHO has H
That gives p(C ) = 0.1244, p(A|C) = 1 and p(C|A) ~= 0.65
All simple. The issue that RC works on is looking at the case of the specific honour:
D = LHO has Qxx, RHO has K
Now if we were to be naive, we might think p(D) = 0.0622, p(A|D) = 1 and p(D|A) ~= 0.32. RC reminds us that this calculation is wrong and that the figure needs to be doubled. That is all it does, it forces non-mathematicians to use the correct interpretation of Bayes' Theorem. There are of course multiple correct ways of arriving at the same answer, as evidenced by the previous cases. But bridge players are usually not so much interested in obtaining the precise probabilities as in knowing the best line of play. And the beauty of RC is in providing that answer without having to know any maths at all. So no, there is no fallacy here. The underlying maths is sound.
#102
Posted 2016-July-18, 12:23
Zelandakh, on 2016-July-18, 07:16, said:
A = LHO has xx, RHO has at least one honour
B = LHO has xx, RHO has KQ
Are you sure that there is no typo there? If definition of LHO is identical in each of cases A and B, are not RHO hands likewise identical between the cases, by definition?
Psyche (pron. sahy-kee): The human soul, spirit or mind (derived, personification thereof, beloved of Eros, Greek myth).
Masterminding (pron. mstr-mnding) tr. v. - Any bid made by bridge player with which partner disagrees.
"Gentlemen, when the barrage lifts." 9th battalion, King's own Yorkshire light infantry,
2000 years earlier: "morituri te salutant"
"I will be with you, whatever". Blair to Bush, precursor to invasion of Iraq
#103
Posted 2016-July-18, 12:38
#104
Posted 2016-July-18, 12:41
Stephen Tu, on 2016-July-18, 12:38, said:
Ah yes, obvious when thinking what we are trying to achieve.
Psyche (pron. sahy-kee): The human soul, spirit or mind (derived, personification thereof, beloved of Eros, Greek myth).
Masterminding (pron. mstr-mnding) tr. v. - Any bid made by bridge player with which partner disagrees.
"Gentlemen, when the barrage lifts." 9th battalion, King's own Yorkshire light infantry,
2000 years earlier: "morituri te salutant"
"I will be with you, whatever". Blair to Bush, precursor to invasion of Iraq
#105
Posted 2016-July-18, 15:42
Stephen Tu, on 2016-July-18, 12:38, said:
Exactly, A is meant to be the situation we find ourselves in at the point of decision - xx-H + H in an unknown hand - and this is necessary to have the correct conditional probability at the end of it.
#106
Posted 2016-July-18, 18:16
1eyedjack, on 2016-July-15, 15:18, said:
Say North has ♠KT987 and 8 non-spades opposite South ♠A654 and 9 non-spades, and no other information. South cashes ♠A to which all follow, East with ♠Q, West low. West also follows low the next Spade. Crunch time.
Now suppose that East is known to play Q from QJ doubleton with probability p where 0<=p<=1 (and therefore J from that holding with probability 1-p)
The question to resolve is what is the value of p such that when the Q appears on the first round, the success rate in playing for the drop on the second round equals that of the finesse.
West has 11 vacant spaces in which to hold the J. East has 12.
Under PRC, East's vacant spaces are effectively reduced by a factor p. If p=0, the finesse is certain to succeed. If p=1, the drop is slight favourite by a factor 12:11. If p=0.5 (ie random case), then the finesse is favourite by the ratio 11:6, as predicted by classic PRC.
The break even point arises when the ratio 11:12p = 1:1
ie when p = 11/12.
I conclude therefore that even substantial deviations from random can be afforded before it upsets the recommended finesse suggested by PRC.
Question: You thought you detected a B-I-T by East. East plays an honor(queen or jack). Do you now play East for the other honor?
#107
Posted 2016-July-18, 19:06
jogs, on 2016-July-18, 18:16, said:
Surely anyone above a rank beginner has thought about which one to play many times, and doesn't need to hesitate at the table. Unless it's an obvious enough hesitation that everyone notices it (and possibly comments on it), I'm going to play the hitch to mean 'I need to think about playing my singleton in tempo so he doesn't work out the honours are split' or for them to be playing games.
#108
Posted 2016-July-20, 08:38
eagles123, on 2016-July-16, 09:26, said:
At the very least if you're determined to give him the benefit of the doubt, move it to another forum - I suggest Novice and Beginner.