The wine/water paradox is an apparent paradox in probability theory. It is stated by Michael Deakin as follows:
The core of the paradox is in finding consistent and justifiable simultaneous prior distributions for
x
1 | |
x |
This calculation is the demonstration of the paradoxical conclusion when making use of the principle of indifference.
To recapitulate, We do not know
x
Now, making use of the principle of indifference, we may assume that
x
x
xt
xmin<xt<xmax
xt
\operatorname{Prob}\{x\lext\}=
xt-xmin | |
xmax-xmin |
=
1 | |
8 |
(3xt-1).
As a function of the threshold value
xt
0
1
Consider the threshold
xt=2
\operatorname{Prob}\{x\le2\}=
1 | |
8 |
(3 ⋅ 2-1)=
5 | |
8 |
Now consider
y=
1 | |
x |
\operatorname{Prob}\{y\geyt\}=
xmax(1-xminyt) | |
xmax-xmin |
=
3 | |
8 |
(3-yt)
This is the function which is
0
1
\tfrac{1}{xmin
Now taking the corresponding threshold (also half as much water as wine). We conclude that
\operatorname{Prob}\left\{y\ge\tfrac{1}{2}\right\}=
3 | |
8 |
3 ⋅ 2-1 | |
2 |
=
15 | |
16 |
=
3 | |
2 |
5 | |
8 |
The second probability always exceeds the first by a factor of . For our example the number is .
Since , we get
5 | |
8 |
=\operatorname{Prob}\{x\le2\}=P*=\operatorname{Prob}\left\{y\ge
1 | |
2 |
\right\}=
15 | |
16 |
>
5 | |
8 |
a contradiction.