MONTY-HALL THEOREM
:
BAYES-PRICE RULE (BAYES THEOREM) FOR A THREE PARAMETER EVENT SPACE
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/
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h ps://o cid.o g/0000-0003-3483-3521
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ABSTRACT
This esea ch epo p esen s he s a emen o he Mon y-Hall Theo em and
p o ides a cons uc i e p oo by sol ing he classical Mon y-Hall P oblem. I
es ablishes he ac ha he p obabili y o winning he p ize is indeed una ec ed by
a swi ched-choice – e y much unlike he mos p e alen and widely accep ed
posi ion held by he Leading Subjec -Ma e -Expe s.
Keywo ds: Mon y-Hall Theo em; Bayes-P ice Rule; Bayes 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
Subjec -Ma e -Expe s, ma hema icians, s a is icians, logicians, 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.
© D (P o ) Kesha a P asad Halemane Page 2 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 2 o 11 Mon y-Hall Theo em
We p esen he s a emen o ou newly o mula ed Mon y-Hall Theo em, and
p o ide a cons uc i e p oo by sol ing he classical Mon y-Hall P oblem. I
es ablishes ha he p obabili y o he gues winning he p ize is indeed 1/2
i espec i e o whe he he gues s ays wi h he ini ial choice o goes o a swi ched
choice a e ga he ing he in o ma ion om he hos who e eals a losing choice.
2. PROBLEM DESCRIPTION - INPUT DATA
Le us conside he so-called 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
an ini ial choice o a doo o claim he p ize; he hos who knows he loca ion o he
p ize as well as he ini ial choice made by he gues , now e eals a dis inc ly
di e en and ye a losing choice, by opening a hi d doo . Then he 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;
(3) le zq {1,2,3} be he doo q opened by he hos z o e eal a losing choice.
Le he symbol ‘ai’ deno e he e en /ac ion [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 a e 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 mus be 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 .
3. ASSUMPTIONS
I is assumed ha he p ize is hidden andomly behind one o he h ee doo s, each
o he e en s [x {1,2,3}] being conside ed equip obable. Also, he ini ial choice
o he doo [yp{1,2,3}] chosen by he gues is also a andom (blind) choice.
The hos knows he doo behind which he p ize is hidden and also he doo ha
is he ini ial choice o he gues . The e o e, 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. Tha is, (zq ≠ yp) & (zq ≠ x ). This dependency o zq
on yp and x does indeed limi he a ailable 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
© D (P o ) Kesha a P asad Halemane Page 3 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 3 o 11 Mon y-Hall Theo em
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 which o he wo doo s o open, 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. He e, i is assumed ha whene e zq ≠ (yp = x ) he hos ’s choice
be ween he wo a ailable op ions is indeed equip obable.
Table-1 lis s he 12 mu ually-exclusi e oge he -exhaus i e possible al e na i es
o he combined- iple-e en space along wi h he ele an ap io i p obabili ies.
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-1: 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
Twel e Mu ually-Exclusi e Toge he -Exhaus i e Al e na i e-Possibili ies
4. MONTY-HALL THEOREM : CLASSICAL MONTY-HALL PROBLEM
Gi en ha he ini ial choice o he gues is, say doo -1 (e en [y1]); and ha he
hos opens he doo , say doo -3 (e en [z3]) o e eal a losing choice, ha is
di e en om he doo behind which he p ize is hidden, and also di e en om he
ini ial choice o he gues ; hen he p obabili y o he gues winning he p ize is gi en
by he apos e io i (condi ional o [z3]) p obabili y o he p ize being hidden behind
he doo -1 (e en [x1]); ha is, P[x1 | z3]. In he case o he classical Mon y-Hall
P oblem, his alue may be compu ed by he applica ion o he Bayes-P ice Rule
(Bayes Theo em) o he case o h ee pa ame e e en (sample)space; and i is
equal o 0.50 - he e o e he op ion o he swi ched choice doesn’ yield any
enhancemen in he chances o winning he p ize.
PROOF
The p oo is simply by sol ing he p oblem, ollowing he below enume a ed s eps.
© D (P o ) Kesha a P asad Halemane Page 4 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 4 o 11 Mon y-Hall Theo em
Fo each equi ed alue, a gene al exp ession is gi en i s ; ollowed by he
classical case.
(1) INPUT DATA
P[x1]; P[x2]; P[x3]; P[y1]; P[y2]; P[y3];
P[z3 | x1y1]; P[z3 | x1y2]; P[z3 | x2y1]; P[z3 | x2y2];
P[z2 | x1y1]; P[z2 | x1y3]; P[z2 | x3y1]; P[z2 | x3y3];
P[z1 | x2y2]; P[z1 | x2y3]; P[z1 | x3y2]; P[z1 | x3y3];
(2) JOINT PROBABILITIES FOR INDEPENDENT EVENTS [x & yp]
P[x1y1] = P[x1]*P[y1]; P[x1y2] = P[x1]*P[y2]; P[x1y3] = P[x1]*P[y3];
P[x2y1] = P[x2]*P[y1]; P[x2y2] = P[x2]*P[y2]; P[x2y3] = P[x2]*P[y3];
P[x3y1] = P[x3]*P[y1]; P[x3y2] = P[x3]*P[y2]; P[x3y3] = P[x3]*P[y3];
(3) VALIDITY CHECK FOR NON-ZERO APRIORI PROBABILITIES
Check and con i m he alidi y o inpu da a alues o applica ion o Bayes-P ice
Rule (Bayes Theo em). The p esence o ze o- alue o any o he ap io i
p obabili ies leading o he in ended condi ional used o de i e he equi ed
apos e io i (condi ional) p obabili ies, can esul in spu ious esul s. App op ia e
al e na i e app oach may be needed in such cases. Fo he classical Mon y-Hall
P oblem, he join p obabili ies lis ed abo e leading o he equi ed condi ionali y
o he hos opening a doo (say z3) will be used in he below calcula ions.
(4) APRIORI PROBABILITY FOR [z3] AS PER THE RULES OF THE GAME
P[z3] = P[z3 | x1y1]*P[x1y1] + P[z3 | x2y1]*P[x2y1] + P[z3 | x1y2]*P[x1y2] + P[z3 | x2y2]*P[x2y2];
= P[x1y1z3] + P[x1y2z3] + P[x2y1z3] + P[x2y2z3];
= 1/18 + 1/9 + 1/9 + 1/18;
= 1/3;
(5) APRIORI CONDITIONAL (w. . . x1) MARGINAL (w. . . yp) PROBABILITY FOR z3
P[z3 | x1] = (P[z3 | x1y1] * P[x1y1] + P[z3 | x1y2] * P[x1y2]) / (P[x1]);
= (P[z3x1y1] + P[z3x1y2] ) / (P[x1]);
= (1/18 + 1/9 ) / (1/3 );
= 1/2;
(6) APOSTERIORI CONDITIONAL (w. . . z3) MARGINAL (w. . . yp) PROBABILITY FOR x1
P[x1 | z3] = (P[z3 | x1] * P[x1]) / (P[z3]);
= (1/2 * 1/3 ) / (1/3 );
= 1/2;
END OF PROOF
© D (P o ) Kesha a P asad Halemane Page 5 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 5 o 11 Mon y-Hall Theo em
5. DISCUSSION
I is o be no ed he e ha he heo em and he p oo uses some speci ic labels o
he doo s, jus o con enience; namely, doo -3 [z3] o he doo opened by he
hos o e eal a losing choice, and doo -1 [y1] o he gues ’s ini ial choice, hus
leading o a decision making p oblem o he gues ha equi es he compu a ion
o he alue P[x1 | z3] and i s complemen a y alue P[x2 | z3]. Howe e , he esul
is nei he es ic ed by no dependen on hese speci ic labels. This is e iden om
he symme y in he da a en ies in Table-1, which shows ha x and yp a e
in e changeable o any gi en zq, and ha he en ies a e iden ical o he h ee
subse s co esponding o each o he alues o zq {1,2,3}. O , i one wishes,
one can e-w i e he heo em in h ee pa s, one co esponding o each o he cases
wi h he hos opening a doo zq {1,2,3}.
Also, one can always conside a scena io whe ein he h ee doo s a e exac ly
iden ical om he iewpoin o he gues , and ha he ini ial choice o he gues is
hen labelled as doo -1, and ha he doo ha is opened by he hos is hen labelled
as doo -3, hus lea ing he emaining doo o be labelled as doo -2. The e o e, i
ge s es ablished ha i espec i e o whiche e be he doo opened by he hos ,
each o he emaining wo doo s ha e equal p obabili y o ha ing he p ize hidden behind i .
6. EARLIER ERRONEOUS RESULT
The Mon y-Hall Theo em ea i ms common-sense based a ional & in ellec ual
easoning, con i med by he esul s ob ained h ough he compu a ions shown in
he p oo . No e ha he Mon y-Hall P oblem is no a p oblem wi h possibly mul iple
co ec solu ions. The e o e, he abo e heo em indi ec ly poin s ou he e oneous
esul ha has been he widely accep ed posi ion by he Leading Subjec Ma e
Expe s who claim ha a swi ched choice has a clea ad an age, wi h he chances
o winning he p ize being 2/3 as agains only 1/3 o s aying wi h he ini ial choice.
The e seems o be a ious app oaches adop ed by he Leading Subjec Ma e
Expe s, o de i e he e y same e oneous esul . Almos all o hem a e
cen e ed a ound he use ( a he he e oneous use) o he ou ap io i
p obabili ies: (1) P[z3x1y1]; (2) P[z3x2y1]; (3) P[z3x1y2]; (4) P[z3x2y2]; leading
o he in ended condi ional [z3] ha is supposed o be used app op ia ely
o de i e he equi ed apos e io i (condi ional) p obabili ies: P[x1 | z3] o be
compa ed wi h P[x2 | z3] in he decision-making p oblem aced by he gues .
Some conside only he wo ap io i e ms (1) & (2) while lea ing ou (e o o
omission) he o he wo e ms (3) & (4) men ioned abo e; as-i ixing [z3y1] as he
condi ionali y a he han [z3]; and de i e he apos e io i p obabili ies: P[x1 | z3y1]
o be compa ed wi h P[x2 | z3y1] - only o ecommend a swi ched choice om [y1]
© D (P o ) Kesha a P asad Halemane Page 6 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 6 o 11 Mon y-Hall Theo em
o [y2] – which in i sel is indeed a se ious Logical Fallacy. This is exac ly
simila o he physical analogy o chasing he p o e bial mi age-wa e s, whe ein
ha pe cep ion i sel anishes, since he e y condi ions ha caused such a
pe cep ion a e iola ed (no mo e alid) by he e y ac ion o mo ing owa ds i .
Some o he s seem o go w ong in hei applica ion o he Bayes-P ice Rule (Bayes
Theo em) - e o o commission - in a si ua ion wi h ze o alue associa ed wi h
ap io i p obabili ies P[y2] & P[y3] - as-i ixing [y1] as a p e-condi ion – an issue o
conce n ha has been clea ly men ioned in he abo e p oo while insis ing on a
check o he alidi y o inpu da a be o e u he p ocessing o de i e apos e io i
(condi ional o [z3]) p obabili ies.
One o he mos s iking e o s is he claim ha he chances o winning by s aying
wi h he ini ial choice is gi en by (P[x1y1z2]+P[x1y1z3]) whe eas he chances o
winning by a swi ched choice is gi en by (P[x1y2z3]+P[x1y3z2]) as-i ixing [x1] as
a p e-condi ion while no aking ad an age o he addi ional knowledge gained om
he hos opening he doo [z3] e ealing a losing choice!
Simila ly, ano he equally in iguing app oach adop ed by some o he s is o
compa e (P[x1y1z3]+P[x2y2z3]) wi h (P[x1y2z3]+P[x2y1z3]) while co ec ly
conside ing [z3] as he apos e io i condi ion al hough no upda ing he equi ed
p obabili ies o e alua ion & compa ison o he wo possible al e na i es [y1] &
[y2] a ailable o he gues !
We a e amazed as o how hese app oaches can be jus i ied by ei he any a ional in ellec ual
easoning o any heo y based on he undamen als o P obabili y & S a is ics. This is indeed an
a ypical case o e oneous ma hema ical o mula ion o he p oblem gi ing ise o an e oneous
model, and/o e en possibly some e oneous p oblem sol ing leading o e oneous esul s, u he
con i med (!?!) by e oneous compu e simula ion e c. in ol ing he Leading Subjec Ma e
Expe s who a e expec ed o wa n us om such misleading possibili ies.
7. 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:
(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 YOUR ecommenda ion?
(4.2) TO SWITCH OR NOT TO SWITCH : THAT IS THE QUESTION!
© D (P o ) Kesha a P asad Halemane Page 7 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 7 o 11 Mon y-Hall Theo em
No e ha you answe mus necessa ily be independen o my ini ial-choice;
al hough Mon y’s choice o opening a doo o e eal a losing choice was dependen
on my ini ial choice which he had o a oid as pe he ules o he game. Hope you
expe ad ice is NEITHER an exempli ica ion o a well-known p o e b “ he g ass
is always g eene on he o he side” NOR any en icemen o chase he p o e bial
mi age-wa e s whe ein ha pe cep ion i sel 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 .
8. 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 he e ms P[z3 | x1 & y1] and P[z3 | x2 & y2] ha a e ully
unde he con ol o he hos . An ex eme si ua ion is when he hos adop s a ce ain 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 apos e io i(condi ional)
ma ginal p obabili ies P[x1 | z3] and P[x2 | z3] 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: One
ex eme case wi h a speci ic s a egy whe ein a swi ched choice has a clea disad an age; and a
second ex eme case wi h a speci ic s a egy whe ein a swi ched choice has a clea ad an age. I
was le ( e e : [12]) as 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. To close his issue once o all, le us p esen
he Mon y-Hall Theo em o he case o s a egis -hos .
9. MONTY-HALL THEOREM : STRATEGIST HOST
The e does no exis any s a egy ha can be adop ed by a s a egis -
hos in he Mon y-Hall P oblem, ha would esul in a si ua ion whe ein
a swi ched-choice will always (i espec i e o he placemen o he p ize
and i espec i e o he ini ial-choice o he gues ) lead o an
enhancemen /diminishmen in he chances o winning he p ize.
The p oo is le o he cool-headed b a e-hea s. I is wo h no ing ha his gene al e sion o he
Mon y-Hall Theo em subsumes he ea lie specialized e sion o he classical case whe ein he
hos andomly chooses be ween he wo a ailable doo s o e eal a losing choice.
No e ha he e a e eigh dis inc ly di e en possible ex eme s a egies ha can be
adop ed by a s a egis -hos in he Mon y-Hall P oblem; co esponding o he h ee
si ua ions ha p o ide an op ion o he hos o open one o he wo a ailable
al e na i e doo s o e eal a losing choice o he gues . Tha is, whene e he ini ial
choice o he gues ma ches wi h he doo behind which he p ize is hidden, he
hos can open one speci ic chosen doo om among he o he wo doo s, each o
which is a losing choice. The e o e, we can iden i y each o hese eigh dis inc
s a egies by a uniquely cha ac e is ic signa u e label {x1y1zu, x2y2z , x3y3zw}
whe e u{2,3}; {3,1}; w{1,2}; o simply by an equi alen label {11u22 33w}.
Re e ing back o Table-1, s a egy S1 co esponds o he scena io ha includes
each o he h ee combined- iple-e en s [x1y1z3] and [x2y2z3] and [x3y3z1] wi h
© D (P o ) Kesha a P asad Halemane Page 8 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 8 o 11 Mon y-Hall Theo em
he join p obabili y o 1/9 o each o hem, whe eas he associa ed h ee
combined- iple-e en s [x1y1z2] and [x2y2z1] and [x3y3z2] a e elimina ed om
conside a ion; while he emaining six combined- iple-e en s [x1y2z3] and
[x2y1z3] and [x1y3z2] and [x3y1z2] and [x2y3z1] and [x3y2z1] emain as such wi h
he join p obabili y o 1/9 o each o hem. I is exac ly he same as de i ing an
Inpu Da a Table S1 o he s a egy S1 by app op ia ely elimina ing he unwan ed
h ee ows om Table-1 and upda ing he join p obabili ies o each o hei
co esponding complemen a y combined- iple-e en s he ein. Simila ly, we can
de i e he Inpu Da a Table o each o he eigh s a egies men ioned abo e, using
which he equi ed compu a ions can be ca ied ou simila o wha is p esen ed in
Sec ion-4 o he p oo o he Mon y-Hall Theo em. The p oo o his heo em (Mon y-Hall
Theo em : S a egis -Hos ) is exac ly simila o ha o he abo e heo em (Mon y-Hall Theo em : Classical Mon y-Hall
P oblem) whe ein each o he eigh s a egies is associa ed wi h hese modi ied Inpu Da a Table en ies.
Table-2 summa izes he esul s o he compu a ions o each o hese eigh
s a egies, gi ing he p obabili y o he p ize being hidden behind one o he wo
doo s co esponding o he case whe ein he hos e eals a losing choice.
Sl.No.
STRATEGY LABEL
P[x1|z3]
P[x2|z3]
P[x2|z1]
P[x3|z1]
P[x1|z2]
P[x3|z2]
S1
{113223331}
1/2
1/2
1/3
2/3
1/2
1/2
S2
{113223332}
1/2
1/2
1/2
1/2
1/3
2/3
S3
{113221331}
2/3
1/3
1/2
1/2
1/2
1/2
S4
{113221332}
2/3
1/3
2/3
1/3
1/3
2/3
S5
{112223331}
1/3
2/3
1/3
2/3
2/3
1/3
S6
{112223332}
1/3
2/3
1/2
1/2
1/2
1/2
S7
{112221331}
1/2
1/2
1/2
1/2
2/3
1/3
S8
{112221332}
1/2
1/2
2/3
1/3
1/2
1/2
Table-2: Eigh Ex eme S a egies -
each wi h h ee pai s o apos e io i p obabili ies o compa ison
The symme y in he esul s as shown in Table-2 abo e is indeed e y in iguing.
No e ha Table-2 p esen s h ee pai s o alues o he compa ison o apos e io i
p obabili ies co esponding o each o he eigh s a egies, hus ha ing a o al o 24 pai s
o alues o compa ison. Fo six o he eigh s a egies, he e a e wo pai s (1/2, 1/2) and
one pai (2/3, 1/3). The wo pai s (1/2, 1/2) indica e he wo scena ios whe ein a swi ched-
choice doesn’ a ec he chances o winning he p ize; whe eas he one pai (2/3, 1/3)
indica es a scena io whe ein a swi ched-choice a ec s he chances o winning he p ize -
an enhancemen om 1/3 o 2/3 o a diminishmen om 2/3 o 1/3 based on he ini ial-
choice o he gues . No e also ha he s a egies S4 & S5, ha e all he h ee pai s wi h
alues (2/3, 1/3) and hence bo h o hem add ess he ques ion posed in Sec ion-8 abo e.
Co esponding o each scena io o an enhancemen he e is a complemen a y
scena io o diminishmen , and hese a e dis ibu ed symme ically among he eigh
dis inc ly di e en ex eme s a egies as can be obse ed om he Table en ies.
© D (P o ) Kesha a P asad Halemane Page 9 o 11 Mon y-Hall Theo em
© D (P o ) Kesha a P asad Halemane Page 9 o 11 Mon y-Hall Theo em
Fo example, in s a egy S1 since P[x2 | z1] is 1/3 and P[x3 | z1] is 2/3 i is clea
ha i he ini ial-choice is doo -2 [y2] hen a swi ched-choice [y3] yields an
enhancemen in he chances o winning he p ize, whe eas i he ini ial-choice is
doo -3 [y3] hen a swi ched-choice [y2] yields a diminishmen in he chances o
winning he p ize.
The e o e, i is es ablished ha he e is no s a egy which p esen s any scena io whe ein a
swi ched-choice always (i espec i e o he placemen o he p ize and i espec i e o he
ini ial-choice o he gues ) yields a clea ad an age o a clea disad an age (enhancemen
o diminishmen ) in he chances o winning he p ize.
10. CONCLUSION
The Mon y-Hall Theo em es ablishes he co ec app oach in o mula ing and sol ing he classical
Mon y-Hall P oblem. I es ablishes he ac ha he p obabili y o winning he p ize is indeed
una ec ed by a swi ched choice.
The mos p e alen and widely accep ed posi ion held by he Leading Subjec Ma e Expe s seems
o ha e a isen om ei he some e oneous p oblem o mula ion gi ing ise o an e oneous
ma hema ical model and/o e oneous p oblem-sol ing app oach, possibly also iddled wi h some
Logical Fallacy, leading o an e oneous esul , ha seems o ha e been jus i ied by some
e oneous compu e simula ion s udies, e c.
The clea ly pa i ioned iple-e en space, wi h he wel e mu ually-exclusi e oge he -exhaus i e
possible al e na i es, as 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.
11. 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
