Difference between revisions of "Principle of Insufficient Reason"
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In the absence of any relevant evidence, agents should distribute their credence equally among all the possible outcomes under consideration. I.e., If there is not enough reason for two quantities to be different (i.e. they're [[Distinguishability|indistinguishable]]), the extrema occurs when the quantities are the same. So basically, a symmetric expression is maximized or minimized when the <math>2</math> quantities are the same. This is what enabled us to say WLOG. | In the absence of any relevant evidence, agents should distribute their credence equally among all the possible outcomes under consideration. I.e., If there is not enough reason for two quantities to be different (i.e. they're [[Distinguishability|indistinguishable]]), the extrema occurs when the quantities are the same. So basically, a symmetric expression is maximized or minimized when the <math>2</math> quantities are the same. This is what enabled us to say WLOG. | ||
==Proof== | ==Proof== | ||
| + | There is not a proof | ||
| + | |||
==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
| − | A hexagon is inscribed in a circle with radius <math>1</math>. What is the maximum area of the hexagon? ([[ | + | A hexagon is inscribed in a circle with radius <math>1</math>. What is the maximum area of the hexagon? ([[Solution to Principle of Insufficient Reason Introductory Problem 1|Solution]]) |
===Intermediate=== | ===Intermediate=== | ||
1. The fraction, | 1. The fraction, | ||
| − | < | + | <cmath>\frac{ab+bc+ac}{(a+b+c)^2}</cmath> |
where <math>a,b</math> and <math>c</math> are side lengths of a triangle, lies in the interval <math>(p,q]</math>, where <math>p</math> and <math>q</math> are rational numbers. Then, <math>p+q</math> can be expressed as <math>\frac{r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>. | where <math>a,b</math> and <math>c</math> are side lengths of a triangle, lies in the interval <math>(p,q]</math>, where <math>p</math> and <math>q</math> are rational numbers. Then, <math>p+q</math> can be expressed as <math>\frac{r}{s}</math>, where <math>r</math> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>. | ||
| − | (Source: I made it. Solution [[ | + | (Source: I made it. Solution [[Problems Collection#Problem3|here]]. See solution 2 for the solution using this method. See solution 1 for the rigorous solution.) |
| − | |||
| − | |||
| − | |||
| + | ===Olympiad=== | ||
| + | <cmath>a+b+c+d+e=8</cmath> | ||
| + | <cmath>a^2+b^2+c^2+d^2+e^2=16</cmath> | ||
| + | Find the maximum value of <math>e</math>. ([[Solution to Principle of Insufficient Reason Introductory Problem 1|Solution 1]] or [[1978 USAMO Problems/Problem 1|Solution 2]]) | ||
{{stub}} | {{stub}} | ||
Latest revision as of 20:42, 6 July 2024
Statement
In the absence of any relevant evidence, agents should distribute their credence equally among all the possible outcomes under consideration. I.e., If there is not enough reason for two quantities to be different (i.e. they're indistinguishable), the extrema occurs when the quantities are the same. So basically, a symmetric expression is maximized or minimized when the 
quantities are the same. This is what enabled us to say WLOG.
Proof
There is not a proof
Problems
Introductory
A hexagon is inscribed in a circle with radius 
. What is the maximum area of the hexagon? (Solution)
Intermediate
1. The fraction,
![\[\frac{ab+bc+ac}{(a+b+c)^2}\]](http://latex.artofproblemsolving.com/6/0/7/60701a0dfd9308e7ddc45f43e5380a0a359f23e1.png)
where 
and 
are side lengths of a triangle, lies in the interval ![$(p,q]$](http://latex.artofproblemsolving.com/c/b/b/cbb6084727fb3e1c6027f7742cc15d4802ad8681.png)
, where 
and 
are rational numbers. Then, 
can be expressed as 
, where 
and 
are relatively prime positive integers. Find 
.
(Source: I made it. Solution here. See solution 2 for the solution using this method. See solution 1 for the rigorous solution.)
Olympiad
![\[a+b+c+d+e=8\]](http://latex.artofproblemsolving.com/6/0/0/60098488436ab00bf6961ec715b867f633984c7c.png)
![\[a^2+b^2+c^2+d^2+e^2=16\]](http://latex.artofproblemsolving.com/b/b/c/bbc4a1f0afb842215438db2a6b30db9dcfcf50f9.png)
Find the maximum value of 
. (Solution 1 or Solution 2)
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