Expected area of a triangle where 1 point is within another triangle

Taniya Melton 2022-10-13 Answered
Expected area of a triangle where 1 point is within another triangle
Suppose we have triangle ABC with area k with a point P chosen inside ABC. What is the expected area of triangle PBC?
I'm pretty sure if we let P be the centroid we get k/3. Also, how would you solve this question for an n-sided polygon?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Marlene Welch
Answered 2022-10-14 Author has 23 answers
Step 1
Let the vertices of the triangle be a, b, c. Then the map
(1) g : ( u , v ) { x ( u , v ) = a 1 + u ( b 1 a 1 ) + v ( c 1 a 1 ) y ( u , v ) = a 2 + u ( b 2 a 2 ) + v ( c 2 a 2 )   ,
maps the arbitrary triangle from
( x , y ) = ( a 1 , a 2 ) , ( b 1 , b 2 ) , ( c 1 , c 2 ) bijectively onto a right triangle with ( u , v ) = ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) (make a sketch for visualization). Therefore the transformation formula for multiple integrals can be used, and we obtain
T f ( x , y ) d ( x , y ) = T f ( x ( u , v ) , y ( u , v ) ) | J g ( u , v ) |   d ( u , v )   .
The Jacobian determinant is a constant: From (1) we obtain
J g ( u , v ) = det [ x u x v y u y v ] = ( b 1 a 1 ) ( c 2 a 2 ) ( c 1 a 1 ) ( b 2 a 2 )   .
Step 2
Therefore we can write
T f ( x , y ) d ( x , y ) = | J g | 0 1 0 1 u f ( x ( u , v ) , y ( u , v ) ) d v   d u   .
The area of an arbitrary triangle a,b,p is f = 1 / 2 base height. In u,v coordinates the base is 1 and the height is v. For normalization the integral has to be divided by the total area that is 1/2. The expected area of a random triangle a,b,p is therefore
(2) E [ A a , b , p ] = 1 2 1 2 | J g | 0 1 0 1 u v d v   d u   = 1 6 | ( b 1 a 1 ) ( c 2 a 2 ) ( c 1 a 1 ) ( b 2 a 2 ) |
As the area of the triangle a, b, c is
A a , b , c = 1 2 | ( b 1 a 1 ) ( c 2 a 2 ) ( c 1 a 1 ) ( b 2 a 2 ) |
the expected area of the random triangle is 1/3 of the area of the original triangle, independently which side is used as a base.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-10-28
How to prove inequality between probabilities of negative-binomial random variable and geometric variable?
Let X be a negative binomial random variable of parameters r, p, then X = Y 1 + Y 2 + + Y r , where Y j , j = 1 , , r are geometric random.variables of parameter p. Show that: P ( X > k ) r P ( Y 1 > k / r )
asked 2022-08-20
If X is a nonnegative σ-subGaussian random variable with P ( X = 0 ) p, what is a good upper bound for P ( X h )?
asked 2022-10-08
Conditional Geometric Probability
A certified public accountant (CPA) has found that nine of ten company audits contain substantial errors. If the CPA audits a series of company accounts, what is the probability that the first account containing substantial errors will occur on or after the third audited account?
The answer key tells me that it should be 0.01. Shouldn't it be 0.991 since P ( Y = 3 ) = 0.009 and 1 0.009 = 0.991?
asked 2022-08-11
Probability about a geometric distribution
If no-one obtains "head", the game continues with the same probabilities as before." If that is the case, why does that affect the probability recursively? Could someone explain why we add the case where no one wins to the probability, and why we multiply it by p?
asked 2022-08-22
Probability given by geometric model: P ( X = k ) = p ( 1 p ) k , calculate P ( X > 1   |   X 2 )
Being X G ( 0.4 ), calculate:
a) P ( X = 3 )
c) P ( X > 1   |   X 2 )
asked 2022-07-23
Probability with elements of geometry
We have a wire of length 20. We bend this wire in random point. And then bend again to get rectangual frame. What's the probability that area of this rectangle is less than 21.
asked 2022-07-21
Probability That the Sides of an Isosceles Triangle is an Equilateral Triangle
The sides of an isosceles triangle are whole numbers, and its perimeter is 30 units. What is the probability that the triangle is equilateral?

New questions