Solution Manual

13

The three door game (Exercise 1.50 in text)

Here,we take up the taskof analyzingtheproblem,e.g.,usingBayes’theorem,to calculatetheposteriorconditionalexpectation.

Therearethreedoors�����������

. Eventsaredeterminedby wherethetreasureis, by which doorwechoose,andby which opensup afterour first selection.I.e.,if we designateby a triple

������������thecollectionof outcomes:treasure in A,

select A, open C which is empty, the event that treasureis behinddoor�

issimply

� ������������������� �������������where

����runoverall possibilities

���

while theeventthatweselect�

is similarly denotedby� � ������������!

Thespaceof outcomesis:" � � ����������#$�%������������$�%����������&�%������'��#$�

�(���������&�%����������)&�%�����������$�%�(�����*���)&��(�'�������&�%���*��+���,$�%�(�'��'���,$�%�(�'��'��#$�

and - thefamily of its subsets

- ���/.0� " �1� �����������$���1� �2�������������31465%!��

Following thedescriptionof theproblem,notall elementaryeventshavethesameprobability. Wehave

78�9 � :+�;� �����������#&�%�����������$�<

� :+�;� �(�+������)&�%����������$�<� :+�;� �(�'��'���,$�%�(�'��'��#�<� :+�;�%��������#�<� :+�;�%�����*��=�<� :+�;�/��������#�<� :+�;�/����'���*�<� :+�;�/�'������=�<� :+�;�/�'������*�<

14

while 78 9

7> � :�� � ����������#�<?�@:+�;� ������������<

� :�� � �(����+���,�<?�@:+�;� �(�+������$�<� :�� � �(�'��*���,$�<A�B:+�;� �(�'��'��#�<&!

The above aredueto the fact that, e.g.,� ������������

containsonly oneelemen-tary outcome,becauseonly

�canbeopened,whereas

� ������������$�containstwo

outcomeswith equalprobability. Thesefactsdefineeverything!Fromhereon we canenumerate,we cancheckindependence,etc.etc.or, we

cancomputetheconditionalprobabilityfor potentialbenefitin a“switchingstrat-egy”. This is whatwedo next.

First,by directenumeration:Given that we select

�, and that the door

�hasbeenopenedwith no treasure

behindit, weneedto compare:+�;� �������������DC�� �2���������$�<

and:+�;� ���*�&������$�DCE� � ��������#$�<$!

If we can show that:+�;� �������������DC�� � ��������#�<GF :+�;� ���*�����&��$�DC�� �2���������#$�<

thena “switching strategy” helps.Usingtheprobabilitiesof theelementaryeventslistedearlier, wecomputethat

:+�;� �������������<H� 78

:+�;� � ��������#�<H� 78 9JI

78 9

7> � 7

K:+�;� �������������MLN� �2���������$�<O� :+�;� ����������#$�<P� 7

Q� :+�;� �������������<R:+�;� �2���������$�<&�

Thus,� �������������

and� �2���������#$�

areobviously independent,and

:+�;� �������������DC�� �2���������$�<S� :�� � �����������$�<:+�;� �2���������#$�<

� 78 !

Similarly (thoughdoneabit differentlyfor variety),

:+�;� ���*�����&��$�DC�� �2���������#$�<S� :+�;� ���*�����&��$�MLT� �2���������$�<:�� � � ��������#$�<

15

� :+�;� ���*������#�<:+�;� � ��������#�<

� UV;WUX

� >8 !

Second,usingBayes’theorem:Weagainenumeratepossibilities,exceptthatwe turn thingsaround,whichsome-timesmakesit easier. We compute:

:+�;� �(�'���������DC�� �2���������$�<S� :�� � � ��������#�DC�� �(�'���������<R:+�;� ���*�����&��$�/:+�;� � ��������#�<

� :�� � � ����������$�DCE� �(�'���������<R:+�;� �(�'�����$��$�<:+�;� � ��������#�<

� UV UVUV;W I UV;W U9

� >8 !

Hereweusedthefactthat� � ��������#�YLT� �(�'���������J��� � ����������$�YLZ� �(�'��������$�

becauseif we selected�

and treasureis in�

,�

is the only choicefor open-ing. This is reflectedin the fact that thereis no eventotherthan

� ���*������#�in� �(�'�����&��$�

.Of course,if werealizethat“no switching”givesusa UV probabilityof winning,

thenwewouldexpectthat“switching” wouldhaveanadvantage,andit does.Butit is notentirelyobvious.Thetoolsandconceptsof probabilityallow asystematicapproachto all suchquestions. The bottom line is that we have to enumeratepossibilities.In thiscaseit maylook tedious,but in many caseswecansavequiteabit of effort usingsuchanapproach(e.g.,usingBayes’theorem).

Incidentally, asyou cansee,it is oftenthecasethat theright “language”andtheright “notation” allowsusto think andcomputemoreeasilythanwithout it!