REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON THE MONTY-HALL PROBLEM

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EXECUTIVE SUMMARY
REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON THE MONTY-HALL PROBLEM
Le x {1,2,3} be he doo behind which he p ize x is hidden. Le yp{1,2,3} be he ini ial choice p o
he gues y. Le z{1,2,3} be he doo q opened by he hos z o show a losing choice. Also, x and yp a e
mu ually independen ; bu zq is dependen on bo h yp and x , ha is, zq ≠ (yp, x ). The symbol ai deno es
he e en [E{(a=i)}] o any ‘agen ’ a{x,y,z} and ‘doo ’ i{ ,p,q}={1,2,3}. The Table lis s he 12
mu ually-exclusi e oge he -exhaus i e possibili ies o he combined- iple-e en [x &yp&zq] :
Sl.No.
[x ]
[yp]
[x &yp]
[zq]
[x &yp&zq]
P[x ]
P[yp]
P[zq  (x & yp)]
P[x &yp&zq]
01
1
1
11
2
112
1/3
1/3
1/2
1/18
02
1
1
11
3
113
1/3
1/3
1/2
1/18
03
1
2
12
3
123
1/3
1/3
1
1/9
04
1
3
13
2
132
1/3
1/3
1
1/9
05
2
1
21
3
213
1/3
1/3
1
1/9
06
2
2
22
1
221
1/3
1/3
1/2
1/18
07
2
2
22
3
223
1/3
1/3
1/2
1/18
08
2
3
23
1
231
1/3
1/3
1
1/9
09
3
1
31
2
312
1/3
1/3
1
1/9
10
3
2
32
1
321
1/3
1/3
1
1/9
11
3
3
33
1
331
1/3
1/3
1/2
1/18
12
3
3
33
2
332
1/3
1/3
1/2
1/18
Table: Twel e combined- iple -e en possibili ies along wi h i s join -p obabili ies.
[x ]: p ize x behind doo ; [yp]: gues y choses doo p; [zq]: hos z e eals doo q
Mu ually-Exclusi e Toge he -Exhaus i e Al e na i e-Possibili ies
COMMENT ON THE APPROACH ADOPTED BY LEADING SUBJECT-MATTER-EXPERTS
One o he a ious app oaches adop ed by he Leading Subjec -Ma e -Expe s, is o conside he wo join
p obabili ies P[x1&y1&z3] = 1/18 and P[x2&y2&z3] = 1/18 and compa e hem wi h he wo join
p obabili ies P[x1&y2&z3] = 1/9 and P[x2&y1&z3] = 1/9; and somehow (w ongly) de i e he p obabili y o
winning wi h ini ial choice o be 1/3 and he p obabili y o winning wi h swi ched choice o be 2/3; wi hou
e en ealizing ha each o hese ou alues is an a-p io i p obabili y co esponding o he ou e en s leading
o he hos e ealing a losing choice; and none o hem co espond o any upda ed a-pos e io i p obabili y
ha inco po a es he knowledge o he losing choice e ealed by he hos .
Also, he e seems o be some Logical Fallacy in he unde lying easoning used he e o .
The only one a-pos e io i condi ionali y is (zq=z3) whe eas (yp=y1) can’ be pinned down as a condi ionali y.
Logical Consis ency equi es ha any condi ionali y used in he e alua ion p ocess canno be li ed a e he
e alua ion p ocess while implemen ing he decision a i ed a based on ha e y condi ionali y. O -Else, an
e oneous p oblem o mula ion & e oneous model na u ally leads o e oneous esul s, ha ge s con i med
h ough some e oneous compu e simula ion s udies, e c.
PROPOSED APPROACH
The gues has wo op ions, o swi ch o no o swi ch. The ma hema ical model mus cap u e he cen al c ux
o he decision-making p ocess; whe ein he gues goes h ough a wo-s ep p ocedu e o a i e a he decision:
(S ep-1) wi hd aw/cancel he ini ial choice o doo -1; (S ep-2) e alua e he equi ed a-pos e io i p obabili ies
based on he knowledge gained om he hos ega ding a losing choice.
Wha is equi ed is o compa e he alues o he wo condi ional (w. . . z3) ma ginal (w. . . yp) p obabili ies,
P[x1  z3] = (P[x1  y1 & z3] + P[x1  y2 & z3]) / P[z3] = (1/18 + 1/9) / (1/3) = 1/2; and
P[x2  z3] = (P[x2  y1 & z3] + P[x2  y2 & z3]) / P[z3] = (1/9 + 1/18) / (1/3) = 1/2;
hus, leading o he ecommenda ion o he gues ha i eally doesn’ ma e ei he way.
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REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON
THE MONTY-HALL PROBLEM
PIPR:©: D .(P o .) Kesha a P asad Halemane,
P o esso - e i ed om
Depa men o Ma hema ical And Compu a ional Sciences
Na ional Ins i u e o Technology Ka na aka, Su a hkal
S ini asnaga , Mangalu u - 575025, India.
SASHESHA, 8-129/12 Sowjanya Road, Naiga a Hills,
Bika naka e, Kulsheka Pos , Mangalu u-575005. Ka na aka S a e, India.
h ps://www.linkedin.com/in/kesha ap asadahalemane/
h ps://colab.ws/ esea che s/R-3D34E-09884-MI42Z
h ps://gi hub.com/KpH8MACS4KREC2NITK
h ps://o cid.o g/0000-0003-3483-3521
h ps://os .io/x 8/
ABSTRACT
This esea ch epo p esen s a deep e-look a he classical Mon y-Hall P oblem, e u ing he
widely accep ed posi ion held by he Leading Subjec -Ma e -Expe s, and es ablishing ha he e
is no a ional basis o a swi ched choice in he decision o be made by he gues o he game show.
Logical consis ency equi es ha any condi ionali y used in he e alua ion p ocess canno be li ed
a e he e alua ion p ocess, implemen ing he decision a i ed a based on ha e y condi ionali y.
Keywo ds: A-P io i P obabili y; A-Pos e io i P obabili y; Mu ually Independen E en s;
Mu ually Exclusi e Toge he Exhaus i e Al e na i es; Join P obabili y;
Res ic ed P obabili y; Ma ginal P obabili y; Condi ional P obabili y;
Mon y-Hall Theo em.
AMS MSC Ma hema ics Subjec Classi ica ion: 60A99; 60C99; 62A99; 62C99.
1. INTRODUCTION
The classical “Mon y-Hall P oblem”, also e e ed o as he “Th ee-Doo P oblem” is based on a
game show “Le ’s Make a Deal” whe ein he hos e eals a losing choice o he gues , who had
ea lie made an ini ial choice, and in u n o e s he gues an en icing op ion o swi ch om he
ini ial choice o a second a ailable choice wi h an aim o enhance he chances o winning he p ize.
The mos p e alen & widely accep ed posi ion, as epo ed in li e a u e, among he leading eminen
ma hema icians, s a is icians, logicians, Subjec -Ma e -Expe s and a ional in ellec uals, is ha an
app op ia e de ailed s udy & analysis o he scena io using he well accep ed s anda d app oach o
P obabili y & S a is ics, would lead o a ecommenda ion o he gues o swi ch o he second
a ailable choice based on he knowledge ob ained om he hos e ealing a losing choice.
I is a gued ha he de aul o s icking o he ini ial choice will esul in a p obabili y o success
being only one- hi d whe eas a swi ch o he al e na i e second a ailable choice will esul in a
p obabili y o success being wo- hi d and hence a swi ched choice is ecommended. Howe e , i
will be shown he e ha his app oach i sel canno be jus i ied, and he e o e he esul an
ecommenda ion o swi ched choice is indeed baseless.
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2. DESCRIPTION OF THE PROBLEM - BACKGROUND SCENARIO
We shall ocus only on he so-called classical Mon y-Hall P oblem, o he pu pose o his epo .
Fo he sake o cla i y, le us conside he s anda d classical Mon y-Hall P oblem as epo ed widely
in he li e a u e - wi h a p ize hidden behind one o he h ee doo s; a gues making a choice o he
doo o pick he p ize; he hos who knows he loca ion o he p ize as well as he choice made by
he gues , now e eals a dis inc ly di e en ye a losing choice. The hos also o e s he gues , an
op ion o swi ch om he ini ial choice o he now a ailable second choice, an icipa ing an enhanced
chance o winning he p ize, based on he knowledge ob ained abou a losing choice.
Le us ep esen he e en s/ac ions associa ed wi h he h ee doo s: (1) le x {1,2,3} be he doo
behind which he p ize x is hidden (2) le yp{1,2,3} be he ini ial choice o he doo p chosen by
he gues y and (3) le zq{1,2,3} be he doo q opened by he hos z o e eal a losing choice. The
symbol ‘ai’ deno es he e en [E{(a=i)}] o any ‘agen ’ a{x,y,z} and ‘doo ’ i{ ,p,q}={1,2,3}.
I is essen ial o no e he e ha x and yp being mu ually independen o each o he as well as
independen o zq, whe eas zq i sel is dependen on bo h x and yp, as pe he ules o he game.
Also, no e ha he ocus is on he decision-making p ocess & he ac ion o be aken by he gues .
So, he p oblem o mula ion (modelling) mus necessa ily be om he iew-poin o he gues .
Since he gues y has absolu ely ze o knowledge abou x he doo behind which he p ize x is
hidden, no assump ions need o be made, e en abou i s possible p obabili y dis ibu ion. Simila ly,
he ini ial choice yp o he doo p chosen by he gues y is based on ze o-knowledge wi hou any
s a egy as such, and he e o e a bes a andom (blind) guess. Howe e , o acili a e a conc e e
analysis o he p oblem scena io and o p o ide a amewo k owa ds a igo ous ma hema ical
model, i may be use ul o make an assump ion ha hese wo e en s/ac ions a e equally p obable
among he h ee a ailable mu ually-exclusi e oge he -exhaus i e possible al e na i es, each
ha ing an a-p io i p obabili y o 1/3 hus adding up o one.
Now, because he wo e en s/ac ions [x {1,2,3}] and [yp{1,2,3}] a e mu ually independen o
each o he , he join p obabili y o he combina ions o hese wo e en s can be ob ained as he
p oduc o he p obabili ies o he wo independen componen e en s. The e o e, he join
p obabili y o each o he combined-duple -e en s P[E{(x {1,2,3}) & (yp{1,2,3})}] is 1/9 and
he sum o al o hese nine join p obabili ies is one.
No e ha he e en /ac ion o he hos z opening doo q, zq{1,2,3} o show a losing choice, is
dependen on bo h yp and x , as pe he ules o he game show; ha is, (zq ≠ yp) & (zq ≠ x ).
Al hough he hos has ull & comple e knowledge o he p oblem scena io, his dependency o zq
on yp and x does indeed limi his op ions. I u ns ou ha when yp ≠ x he hos doesn’ ha e any
op ion excep o u n o he one and only one emaining doo zq ≠ (yp ≠ x ); whe eas when yp = x
he hos has he op ion o choosing be ween he wo doo s, ha is, zq ≠ (yp = x ). Because he hos
has his op ion, a leas in a es ic ed sense, o choosing his doo zq, i in oduces an unce ain y
o he gues o p edic /expec /an icipa e he hos ’s decision/ac ion in his ega d. Howe e , as
ea lie , o he e y same easons as s a ed abo e, i may be use ul o make an assump ion ha he
hos ’s choice be ween he wo op ions, whene e a ailable, in a es ic ed sense, is equi-p obable
be ween he wo a ailable mu ually-exclusi e oge he -exhaus i e possible al e na i es, each
ha ing a es ic ed p obabili y o 1/2 hus adding up o one.
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3. PROBLEM FORMULATION
Wi h he abo e unde s anding o he backg ound scena io o he classical Mon y-Hall P oblem,
one can de i e ha he e a e exac ly 12 possibili ies o he combined- iple -e en [x &yp&zq] as
ep esen ed in he Table, lis ing each o he 12 iple s along wi h he associa ed join p obabili ies.
No e ha he e en space is o size 12 and no 27 which would ha e been he case i each o he
h ee componen -e en s we e indeed mu ually independen . The i s wo a e independen gi ing
ise o a combined-duple -e en space E{[x &yp]} o size nine. When i is hen combined wi h he
hi d componen -e en [._.zq], he e esul s a spli ing, in h ee cases. In he h ee cases whe e
[x ]=[yp], ha is, [x1y1.], [x2y2.], [x3y3.] he hi d componen -e en [._.zq] ge s wo al e na i e
possibili ies; [zq]{[z2]˅[z3]} and [zq]{[z1]˅[z3]} and [zq]{[z1]˅[z2]} espec i ely. Whe eas
in he o he six cases [x1y2.], [x1y3.], [x2y1.], [x2y3.], [x3y1.], [x3y2.] whe e [x ]≠[yp], he hi d
componen -e en [._.zq] has a ixed choice since he e is one and only one single possibili y
sa is ying he game equi emen {[zq]≠([yp]≠[x ])}; no spli ing in o mul iple al e na i e op ions.
Sl.No.
[x ]
[yp]
[x &yp]
[zq]
[x &yp&zq]
P[x ]
P[yp]
P[zq  (x & yp)]
P[x &yp&zq]
01
1
1
11
2
112
1/3
1/3
1/2
1/18
02
1
1
11
3
113
1/3
1/3
1/2
1/18
03
1
2
12
3
123
1/3
1/3
1
1/9
04
1
3
13
2
132
1/3
1/3
1
1/9
05
2
1
21
3
213
1/3
1/3
1
1/9
06
2
2
22
1
221
1/3
1/3
1/2
1/18
07
2
2
22
3
223
1/3
1/3
1/2
1/18
08
2
3
23
1
231
1/3
1/3
1
1/9
09
3
1
31
2
312
1/3
1/3
1
1/9
10
3
2
32
1
321
1/3
1/3
1
1/9
11
3
3
33
1
331
1/3
1/3
1/2
1/18
12
3
3
33
2
332
1/3
1/3
1/2
1/18
Table: Twel e combined- iple -e en possibili ies along wi h i s join -p obabili ies.
[x ]: p ize x behind doo ; [yp]: gues y choses doo p; [zq]: hos z e eals doo q
Mu ually-Exclusi e Toge he -Exhaus i e Al e na i e-Possibili ies
4. GENERAL ANALYSIS OF THE DECISION-MAKING SCENARIO
In his sec ion, a gene al analysis o he decision-making scena io (modelling, om he gues ’s
iewpoin ) is p esen ed i s , wi hou being cons ained by he h ee assump ions men ioned ea lie .
The combined- iple-e en space ep esen ed by E{[x & yp & zq]} is pa i ioned in o 12 mu ually
exclusi e oge he exhaus i e possible a ailable al e na i es – al hough wi h no assump ions abou
he p obabili y dis ibu ions, jus o accommoda e o possible speci ic scena ios, especially i &
when someone wishes o y ou compu e simula ion s udies, e c. Speci ic esul s pe aining o he
da a en ies gi en in he Table, may always be easily wo ked ou by plugging he co esponding
da a alues o each o he conce ned pa ame e s as needed.
The decision/choice/ac ion o he hos , ep esen ed by zq{1,2,3} being dependen on x {1,2,3}
and yp{1,2,3}; implying, ha he join p obabili y o he combined- iple -e en e e ed he ein
be de e mined by he co esponding condi ional p obabili y:
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Tha is, in gene al, o any [zq] and [x ] and [yp] we ha e,
P[zq & x & yp] = P[zq  x & yp] * P[x & yp]; (Eqn.1)
F om he ules o he game, when [zq] ≠ {[yp] ≠ [x ]} we ha e he condi ional p obabili y,
P[zq  x & yp] = 1; (Eqn.2)
and he e o e, we ge he join p obabili y,
P[zq & x & yp] = P[x & yp]; (Eqn.3)
whe eas, when [zq] ≠ {[yp] = [x ]} we canno make any s onge s a emen , excep he gene al
condi ion o he es ic ed (condi ional) p obabili y:
0 ≤ P[zq  x & yp] ≤ 1; (Eqn.4)
and he e o e, we ge he join p obabili y, as pe Eqn.1 abo e,
P[zq & x & yp] = P[zq  x & yp] * P[x & yp]; (Eqn.5)
The en ies in he Table ha e been illed based on he compu a ions as in Eqn.1 o 5 abo e.
Wi h his in o ma ion, we de e mine he a-pos e io i (condi ional on zq) ma ginal-p obabili y o
he p ize being hidden behind doo (x= ), as ollows;
P[x  zq] * P[zq] = P[zq  x & yp] * P[x & yp] + P[zq  x & y ] * P[x & y ] (Eqn.6)
Tha is, P[x  zq] * P[zq] = P[x & yp & zq] + P[x & y & zq] (Eqn.7)
No e ha Eqn.6 is he co ec applica ion o he Bayes-P ice Rule o equi alen ly Eqn.7 he co ec
me hod o combining he join p obabili ies o de e mine he ma ginal p obabili y.
Using Eqn.7 in speci ic ins ances, o example, wi h (zq=z3) as:
P[x1  z3]*P[z3] = P[z3  x1 & y2] + P[z3  x1 & y1]; (Eqn.8)
P[x2  z3]*P[z3] = P[z3  x2 & y1] + P[z3  x2 & y2]; (Eqn.9)
No ice in Eqn.8 & Eqn.9 abo e, ha he six e ms P[x1y2z3], P[x2y1z3], P[x1y3z2], P[x3y1z2],
P[x2y3z1], P[x3y2z1] a e no unde he con ol o he hos , as can be con i med om Eqn.2 &
Eqn.3 abo e; whe eas, he o he six e ms P[x1y1z3], P[x2y2z3], P[x1y1z2], P[x3y3z2],
P[x2y2z1], P[x3y3z1] a e indeed unde he di ec con ol o he hos (o cou se, wi hin ce ain
limi s as pe Eqn.4 o he es ic ed p obabili y) based on wha e e s a egy ha he hos decides
and ac s acco dingly, while ollowing he ules o he game.
The decision o he gues as o whe he o a ail he o e o he hos o op o a swi ched choice,
say, om doo -1 o doo -2 a e knowing he losing choice behind doo -3 as e ealed by he hos ,
mus be based on a compa ison be ween he wo a-pos e io i(condi ional) ma ginal p obabili ies
P[x1  z3] gi en by Eqn.8 and P[x2  z3] gi en by Eqn.9 as shown abo e. Howe e , any speci ic
answe needs o be de i ed based on he ela i e magni udes o he ou join p obabili ies in ol ed
he ein, namely, P[x1y1z3], P[x1y2z3], P[x2y1z3] and P[x2y2z3]. Tha is whe e he need a ises
o pin down ce ain unce ain ies (a leas he ones ha a e no unde he con ol o he hos ) by
assuming ce ain p obabili y dis ibu ion, as o example, P[x ] and also P[yp] o be uni o mly
dis ibu ed among he a ailable (in his case, h ee) al e na i es. I he decision & ac ion o he
hos can also be assumed o adhe e o ce ain p obabili y dis ibu ion o ce ain s a egy, i can be
used o make speci ic compa isons ha will lead o a i m ecommenda ion o he gues as o
whe he i is wo h a all o conside a swi ched choice. The Table en ies assume ha he hos
adhe es and ollows a uni o m dis ibu ion, in he sense ha whene e aced wi h wo/mul iple
al e na i es, he speci ic choice o any one o hem is equally-p obable and oge he -exhaus i e.

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5. WHAT IS WRONG WITH THE EXISTING APPROACH
Wha is equi ed is a compa ison be ween he wo alues o he condi ional (w. . . zq) ma ginal
(w. . . yp) p obabili ies as can be de i ed using Eqn.6 & Eqn.7, o Eqn.8 & Eqn.9 abo e, ha is,
P[x1  z3] =(1/18 + 1/9)/(1/3)=1/2; and P[x2  z3] =(1/9 + 1/18)/(1/3)=1/2; (Eqn.10)
hus, leading o he ecommenda ion ha i eally doesn’ ma e ei he way.
One o he a ious app oaches adop ed by he Leading Subjec -Ma e -Expe s, is o conside he
wo join p obabili ies P113 = P[x1y1z3] = 1/18 and P223 = P[x2y2z3] = 1/18 and compa e hem
wi h he wo join p obabili ies P123 = P[x1y2z3] = 1/9 and P213 = P[x2y1z3] = 1/9; and somehow
(e o o commission in he w ong applica ion o he Bayes-P ice Rule) de i e he p obabili y o
winning wi h ini ial choice o be 1/3 and he p obabili y o winning wi h swi ched choice o be 2/3;
wi hou e en ealizing ha each o hese ou alues is an a-p io i p obabili y co esponding o he
ou e en s leading o he hos e ealing a losing choice; none o hem co espond o any upda ed
a-pos e io i p obabili y ha inco po a es he knowledge o he losing choice e ealed by he hos .
Also, he e seems o be some Logical Fallacy in he unde lying easoning used he e o . No e ha
he only one a-pos e io i condi ionali y is (zq=z3) whe eas (yp=y1) can’ be pinned down as a
condi ionali y. Logical Consis ency equi es ha any condi ionali y used in he e alua ion p ocess
canno be li ed a e he e alua ion p ocess while implemen ing he decision a i ed a based on
ha e y condi ionali y. O -Else, an e oneous p oblem o mula ion & e oneous model na u ally
leads o e oneous esul s, ha ge s con i med h ough some e oneous compu e simula ion
s udies, e c.
Ano he app oach aken by he Leading Subjec -Ma e -Expe s seems o be based on an e oneous
(e o o omission) compa ison be ween he alues o he wo join p obabili ies –
P[x1 & y1 & z3] = 1/18 and P[x2 & y1 & z3] = 1/9; (Eqn.11)
which hey seem o somehow w ongly combine oge he o ge he wo condi ional p obabili ies
P[x1  z3] = 1/3 and P[x2  z3] = 2/3; (Eqn.12)
and hen li he condi ionali y (yp=y1);
hus, leading o he ecommenda ion o he gues o swi ching o e o doo -2 ( ha is, yp=y2).
Again, no e ha he only one a-pos e io i condi ionali y is (zq=z3) whe eas (yp=y1) canno be
pinned down as an a-pos e io i condi ionali y, since his e y ini ial choice o he gues is indeed
unde e-e alua ion and hence subjec o change.
I is indeed e y in iguing o no e ha he peculia an i-symme y in he da a as lis ed in he Table,
P[x2y2z3] = P[x1y1z3] ≤ P[x1y2z3] = P[x2y1z3] (Eqn.13)
causes he e y same coun e -in ui i ely pa adoxical en icemen exis s o a swi ched choice,
wha e e migh ha e been he ini ial choice – jus i ying ha MHP is indeed a pa adox – exac ly
simila o he physical analogy o a mi age, whe ein he pe cep ion anishes, since he e y
condi ions ha caused such a pe cep ion a e iola ed by he e y ac ion o mo ing owa ds i .
The ma hema ical model mus cap u e he cen al c ux o he decision-making p ocess; whe ein he
gues goes h ough an e ec i ely wo-s ep p ocedu e o a i e a he decision;
(S ep-1) wi hd aw/cancel he ini ial choice o doo -1; ollowed by
(S ep-2) e-e alua e he equi ed a-pos e io i p obabili ies based on he knowledge gained om
he hos ega ding a losing choice; and implemen he decision by aking ac ion acco dingly.
The clea ly pa i ioned iple-e en space, wi h 12 mu ually-exclusi e and oge he -exhaus i e
possible al e na i es, ep esen ed in he Table, is a ail-sa e amewo k o s udy, analyze & sol e
he p oblem – no possibili y o missing any ele an (and/o including any i ele an ) componen
e ms while going h ough he equi ed calcula ions in o de o de i e wha e e desi ed esul s.
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PIPR:©: D (P o )Kesha a.P asad.Halemane Page 7 o 11 Mon y-Hall-P oblem
6. A CHALLENGE TO THE LEADING SUBJECT MATTER EXPERTS
Le us eph ase he Mon y-Hall P oblem, now ado ned wi h a jewel-on- he-c own as below:
MONTY-HALL-PROBLEM (MHP) TO-SWITCH-OR-NOT-TO-SWITCH : THAT IS THE QUESTION
(1.1) The p ize is hidden behind one o he h ee doo s.
(1.2) I he gues make an ini ial choice o which doo i could be, say doo -1, o claim my p ize.
(1.3) Then Mon y he hos opens a di e en doo , say doo -3, e ealing a losing choice.
(2.1) I am gi en an op ion o wi hd aw/cancel he ea lie choice o doo -1 and swi ch o doo -2.
(2.2) I app ecia e he knowledge o a losing choice and also Mon y’s o e o he op ion o swi ch.
(3.1) I g ab Mon y’s o e , wi hd aw/cancel my ea lie choice o doo -1.
(3.2) Then I e-e alua e he wo choices a ailable o me now, namely doo -1 o doo -2.
(3.3) I ind ha he chances o winning a e exac ly he same be ween he wo a ailable choices;
(4.1) Now ha YOU en e he Hall, I seek YOUR ecommenda ion. Wha is YOU ecommenda ion?
(4.2) TO SWITCH OR NOT TO SWITCH : THAT IS THE QUESTION!
No e ha you answe mus necessa ily be independen o my ini ial-choice (doo -1); al hough
Mon y’s choice o doo -3 e ealing a losing choice was dependen on my ini ial choice (doo -1)
which he had o a oid as pe he ules o he game. Hope you expe ad ice is no an
exempli ica ion o he p o e b “ he g ass is always g eene on he o he side”!
7. COOL-HEADED BRAVE-HEARTS PLAY WITH STRATEGIST HOST
This is somewha a om he so-called classical e sion o he Mon y-Hall P oblem, whe ein we
allow he hos o exe cise wha e e ‘s a egic game-playing’ ha one wishes o play wi h he gues .
The si ua ion can be cap u ed by Eqn.8 & Eqn.9 abo e; whe ein he e ms P[z3  x1 & y1] in Eqn.8
and P[z3  x2 & y2] in Eqn.9 a e ully unde he con ol o he hos . An ex eme si ua ion is when
he hos adop s wha e e s a egy ha pulls down one o hem o ze o and pushes he o he one o
i s maximum alue o he es ic ed p obabili y, namely 1/9. Then i u ns ou ha he alues o he
wo a-pos e io i(condi ional) ma ginal p obabili ies P[x1  z3] ob ained om Eqn.8 and P[x2  z3]
ob ained om Eqn.9 can’ be he same anymo e; in he ex eme case, one will be 1/3 and he o he
will be 2/3; which hen may lead o he wo possibili ies: A i s speci ic s a egy a swi ched choice
has an clea disad an age and a second speci ic s a egy whe ein a swi ched choice has an clea
ad an age. This opic is beyond he scope o he p esen pape . I is an exe cise o he cool-headed
b a e-hea s o igu e ou he wo speci ic s a egies ha would lead o such ex eme si ua ions.
8. CONCLUSION
This esea ch epo p esen s a no el in iguing analysis o he Mon y-Hall P oblem, e u ing he
mos widely accep ed posi ion held by he Leading-Subjec -Ma e -Expe s - and ad oca ing
agains ac ing on any en icing o e s made by he hos o he gues o an op ional swi ch om he
al eady selec ed ini ial choice o a dis inc al e na i e a ailable second choice - why, because he e
is no ad an age gained by op ing o such swi ched choice, in e ms o any enhanced chances o
win he p ize, unlike wha e e has been widely accep ed ill oday.
Logical consis ency equi es ha any condi ionali y used in he e alua ion p ocess canno be li ed
a e he e alua ion p ocess, implemen ing he decision a i ed a based on ha e y condi ionali y,
which seems o has been iola ed by he Leading Subjec Ma e Expe s.
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PIPR:©: D (P o )Kesha a.P asad.Halemane Page 8 o 11 Mon y-Hall-P oblem
The app oach aken by he Leading Subjec -Ma e -Expe s seems o be based on an e oneous
ma hema ical o mula ion o he p oblem, leading o an e oneous model which he e o e yields
e oneous esul s, possibly u he con i med (!?!) by e oneous compu e simula ion s udies e c.
The e o can also be conside ed as ei he an e o o commission, ha is, a w ong
applica ion o he Bayes-P ice Rule, o an e o o omission, ha is, conside ing only he
apos e io i(condi ionalz3)join (x1y1z3 as agains x1y2z3)p obabili ies a he han he equi ed
apos e io i(condi ionalz3)ma ginal(x1y1 xo x1y2 as agains x2y1 xo x2y2)p obabili ies.
No e ha any addi ional knowledge gained, e ealing a losing (undesi able) possibili y, al hough
may lead o an upda ed/smalle sample-space, may no and/o need no necessa ily be speci ic
enough o a e inemen /upda e in he ela i e dis inc ion be ween/among he a-pos e io i
p obabili ies o he upda ed/now-a ailable al e na i es in he esul an upda ed sample-space.
9. RECOMMENDED READING
[1]. Wikipedia Page - h ps://en.wikipedia.o g/wiki/Mon y_Hall_p oblem
[2]. Jason Rosenhouse; “The Mon y Hall P oblem:
The Rema kable S o y o Ma h’s Mos Con en ious B ain Tease ”;
Ox o d Uni e si y P ess, ISBN 978-0-19-536789-8, 2009.
[3]. Jason Rosenhouse; “Games- o -You -Mind_His o y-&-Fu u e-o -Logic-Puzzles”;
P ince on Uni e si y P ess, 2020.
[4]. An hony B. Mo on;
“P ize insigh s in p obabili y, and one goa o a ecycled e o ”;
A xi :1011.3400 2 2010.
[5]. Ma hew A. Ca l on;
“Pedig ees, P izes, and P isone s: The Misuse o Condi ional P obabili y”;
Jou nal o S a is ics Educa ion Volume 13, Numbe 2 (2005);
ww2.ams a .o g/publica ions/jse/ 13n2/ca l on.h ml
[6]. Richa d D. Gill; “The Mon y Hall P oblem is no a P obabili y Puzzle :
I 's a challenge in ma hema ical modelling”;
a Xi :1002.0651 4 2023.
[7]. To s en Enßlin and Ma g e Wes e kamp;
“The a ionali y o i a ionali y in he Mon y Hall p oblem”;
a Xi :1804.04948 4 2018.
[8]. A.P. Fli ney_, D. Abbo ; “Quan um e sion o he Mon y Hall p oblem”;
a Xi :quan -ph/0109035 3 2024.
[9]. Je ey S. Rosen hal; “Mon y Hall, Mon y Fall, Mon y C awl”;
p obabili y.ca/je /w i ing_mon y all
[10]. Ch is ophe A. Pynes; “IF MONTY HALL FALLS OR CRAWLS”;
EuJAP Vol.9, No.2, pp 33-47; 2013.
MHPPIPR:©: D (P o )Kesha a.P asad.Halemane Page 9 o 11 Mon y-Hall-P oblem
PIPR:©: D (P o )Kesha a.P asad.Halemane Page 9 o 11 Mon y-Hall-P oblem
[11]. And ew Vazsonyi, Fea u e Edi o ; “Which Doo Has he Cadillac?”;
The Real-Li e Ad en u es o a Decision Scien is – ea u ed column
www.decisionsciences.o g/DecisionLine/Vol30/30_1/ azs30_1.pd
[12]. Halemane, K.P. (2014);
“Unbelie able O(L1.5) wo s case compu a ional complexi y achie ed by
spdspds algo i hm o linea p og amming p oblem”;
a xi :1405.6902 2025.
10. ACKNOWLEDGEMENT
Le us acknowledge ha , looking back, i seems as i Sa an cas an enchan ingly deep spell o e
F equen is s who in u n pushed P obabilis s o mis ake Bayes-P ice, paying he p ice h ough
e o s o commission and/o e o s o omission, iddled wi h Logical Fallacy. We need o come
ou o ha long d awn in ellec ual hibe na ion o o e six decades, and wake up o eali y.
I mus necessa ily con ess he e ha he co e idea behind his analysis is so s unningly & elusi ely
simple, ha one may simply be aken aback in a p o ound wonde -s uck jaw-d op-silence, maybe
wi h an a e - hough : "oh my goodness, how could i be ha i ne e lashed on me any ime ea lie "!
as was also he case in an ea lie esea ch wo k epo ed in [12] by his au ho .
On his mos auspicious idyaa( ijaya)daSami day [2025OCT02] I was blessed wi h he
ision o o mula e he Mon y-Hall Theo em and i s p oo - as a concise & p ecise app oach
- he p e e ed s yle o p esen a ion especially o he a ge audience consis ing o Ma hema icians,
S a is icians, Logicians, and o he eminen scien is s e c.
11. DEDICATION
To my ಅಜ್ಜ(ajja) Ka inja Halemane Kesha a Bha & ಅಜ್ಜಜ(ajji) Thi umaleshwa i, ಅಪ್ಪ(appa) Shama Bha &
ಅಮ್ಮ(amma) Thi umaleshwa i, o hei eachings h ough lo e, ha quali y ma e s mo e han quan i y;
o my wi e Vijayalakshmi o he e e consis en lo e & suppo ; o my daugh e S iwidya.Bha a i and my
win sons S iwidya.Ramana & S iwidya.P awina o hei lo e & a ec ion.
Whe eas his O iginal Au ho -C ea o holds he (PIPR:©:) Pe pe ual In ellec ual P ope y Righ s,
i is bu na u al ha his legal hei s ( h ee child en men ioned abo e) may a ail he same o pe pe ui y.
To all he cool-headed b a e-hea s, eage ly awai ed bu p obably ye o be isible among he wo ld
p o essionals, especially he Subjec -Ma e -Expe s, who would be a ac ed o and ce ainly capable o
e ec i ely unde s anding wi hou any p ejudice and app ecia ing he deepe insigh s ensh ined in his sho
esea ch epo , who may op o inna e a ional-&-in ellec ual common-sense and simple c ea i i y o e any
sophis ica ed and/o complex heo y in p oblem-sol ing o esol e any seemingly pa adoxical scena io.
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