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The real numbers x and y are chosen at random from the interval [0,1], as in x and y are greater than 0 and less than 1. Find the expected value of x+y.

 

Four points P, Q, R and S are chosen at random on the circumference of a circle. Find the probability that chord PQ and chord RS intersect.

 

Point Q is chosen at random inside equilateral triangle XYZ. Find the probability that Q is closer to the center of the triangle than to X, Y or Z. (In other words, let O be the center of the triangle. Find the probability that OQ is shorter than all of QX, QY and QZ)

 

Right triangle XYZ has legs of length XY=12 and YZ=6. Point D is chosen at random within the triangle XYZ. What is the probability that the area of triangle XYD is at most 12?

 

I really don't know how to even start on these. I'm really confused about the topic expected value. 

 Jan 8, 2020
 #1
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"Right triangle XYZ has legs of length XY=12 and YZ=6. Point D is chosen at random within the triangle XYZ. What is the probability that the area of triangle XYD is at most 12?"

 

The probability is 2/5.

 Jan 8, 2020
 #2
avatar+106519 
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Point Q is chosen at random inside equilateral triangle XYZ. Find the probability that Q is closer to the center of the triangle than to X, Y or Z. (In other words, let O be the center of the triangle. Find the probability that OQ is shorter than all of QX, QY and QZ)

 

See  the following image ;

 

 

We have an equilateral  triangle with a side of 2  and a height of sqrt (3)

O is the point of intersection of the angle bisectors  of the triangle and BY = BO = DO

So.....BY  is 1/3  of the height of  triangle  XYZ

And triangle AYC  is similar to triangle XYZ......so the area of  triangle AYC  =  (1/3)^2  = (1/9) that of triangle XYZ

And Q will be closer to Y than it is to O  when Q is located inside of triangle AYC

 

So.....using symmetry, we will have three such smaller triangles, so their total area = 3(1/9)   = 1/3 that of triangle XYZ

So......the probability that  Q will be closer to O  than to any of the  vertices of triangle XYZ   =  2/3

 

 

cool cool cool

 Jan 8, 2020
 #3
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hint for 1: Keep in mind that you have two variables, each of which is chosen randomly from [0, 1].

 Jan 9, 2020

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