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therefore, the statement claimed by the theorem follows easily from the above two
equations.
Despite the simplicity of this proof, what the theorem asserts is an inversion of
conditions. In fact,
P
(
|
)
A
B
can be computed by means of the inverse conditional
(
|
)
probability
P
. This is a very subtle point. Assume that a test
T
for a disease
D
is wrong only in one out to 1000 cases. Assume that a person is positive to this
test. What is the probability of being affected by D? Is it 0
B
A
.
999? Fortunately it is not
the case. In fact, this deduction confuses
P
(
D
|
T
)
(the probability of the disease D,
when the test T is positive) with
P
(the probability test T positivity, when the
disease D is present). The right way to reason is the application of Bayes' theorem.
For this, assume to know the simple probabilities of
D
and
T
.
D
affects one out of
10000 persons, and
T
provides positivity one out of 1000 cases. Let us assume also
that
T
is very reliable, with
P
(
T
|
D
)
1, that is, if
D
is present, then
T
is positive
with probability 1. This means that in 10000 persons we have 11 positive cases:
10 because
T
is wrong 1 to 1000) plus 1, because 1 to 10000 has the disease
D
and
T
discovers it. In conclusion, the probability of having
D
is less than 10% . In
fact:
(
T
|
D
)=
P
(
D
|
T
)=
P
(
D
)
P
(
T
|
D
)
/
P
(
T
)=
1
/
10000
×
1
/
(
11
/
10000
)=
1
/
11
.
A famous debate about probability is the
three doors quiz
also known as the
Monty
Hall problem
, by the name of the TV conductor in a popular American show. There
are three doors
A
B
,and
C
. A treasure is behind only one of them, and you are
asked to guess which is the lucky door. Let us assume that you choose the door
C
.
Then, another chance is given to you. In fact, they open the door
B
, showing that it
is not the fortunate door, and you are free of confirming your preceding choice
C
,
or changing by guessing
A
. The question is:
What is the more rational choice? Con-
firming C or changing it with A?
Many famous mathematicians concluded that there
is no reason for changing (Paul Erd os, the number one of 20th mathematicians was
among them). However, by using Bayes' theorem it can be shown that changing the
door is the better choice, because it provides a passage from a probability 1
,
/
3ofsuc-
cess to a probability of 2
3. This fact was even confirmed by computer simulations
of the game. We do not present the detailed analysis based on Bayes' theorem, but
we give an intuitive, very convincing reason. If the doors are 100:
D
1
,
/
D
2
,...
D
100
and after you guess
D
100
, the doors
D
2
,
are shown to be unlucky, are you
sure to confirm
D
100
, or rather, are you inclined to believe that is wiser change your
initial choice? In fact, what is crucial in this game is that opening doors change the
the space of events.
D
3
,...
7.6.2
Statistical Inference
The simplest probabilistic schema is an urn of
n
balls, where
m
<
n
are white and
n
m
are black. This model, due to Jakob Bernoulli, assumes a boolean random
variable (the color of the extracted ball), but it is of crucial importance in analyz-
ing and developing general probabilistic concepts. Moreover, it is surprising that
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