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(Kira)

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The optimal auction?

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he Law of Cosines is a theorem which relates the side-lengths and angles of a triangle. It can be derived in several different ways, the most common of which are listed in the "proofs" section below. It can be used to derive the third side given two sides and the included angle. All triangles with two sides and an include angle are congruent by the Side-Angle-Side congruence postulate.

For a triangle with edges of length a, b and c, opposite angles of measure A, B  and C , respectively, the Law of Cosines states:
c² =  a2 + b² -2ab cosC
b² = a²+ c² -2ac cosB
a² = b² + c² -2bc cos A

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Proof:

1. P(A | B) = P(A & B)/P(B) [By definition]

2. P(B | A) = P(A & B)/P(A) [By definition]

3. P(B | A) P(A) = P(A & B) [Multiply 2.) by P(A)]

4. P(A | B) P(B) = P(B | A) P(A) [Substitute 1.) in 3.)]

5. P(A | B) = P(B | A) P(A)/P(B) [Bayes’ Rule]

Bayes' rule is a way of calculating conditional probabilities, probabilities whose value depends on the value of another probability.
It can be used by clinicians in diagnostic conditions to find the accuracy of a test for a disease, given the patient's symptoms ,medical history, as well as the base rate of the disease.
Base rate: odds of person getting disease, e.g. 1/1000 people has a chance of catching a disease.

Winston

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Basically is the inequality between the diameter of a set of points in a Euclidean Space and the radius of the minimum enclosing ball of the set.

Most common is the case of Jung's theorem in the plane, that is n = 2. In this case the theorem states that there exists a circle enclosing all points whose radius satisfies

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A problem about probability.
3 doors, 1 car, and 2 goats. You get the choice to switch after 1 door is revealed to be a goat. Do you switch?

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Banache-Tarski theorem... 2 halves are more than a whole

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Banache-Tarski theorem... 2 halves are more than a whole

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A clever way to cut a diamond in half and have two equally large diamonds as a result!!