What is the probability for x to be

Siliberti7cd

Siliberti7cd

Answered question

2022-03-15

What is the probability for x to be positive only?
If x, y, zR and x+y+z=5,,xy+yz+zx=3, what is the probability that x>0?
a) 316
b) 516
c) 1316
d) 1516

Answer & Explanation

orangepaperiz7

orangepaperiz7

Beginner2022-03-16Added 9 answers

The solutions of your two equations form a circle in xyz space, the intersection of the sphere x2+y2+z2=19 with the plane x+y+z=5, which can be parametrized as

x = 5 3 8 3 cos ( t ) y = 5 3 + 4 3 sin ( t ) + 4 3 cos ( t ) z = 5 3 4 3 sin ( t ) + 4 3 cos ( t )

It's not at all clear how to interpret "probability" in this context, but there is no t for which x>0 while y0 and z0. So if x>0 only means x>0 while y0 and z0, the answer must be 0. On the other hand, if you asked for the probability that x>0, one possible interpretation would be the probability that x>0 if you took a random t uniformly from [π,π]. Then you'd get x>0cos(t)<58, i.e. with probability
1arccos(58)π0.7149010415. This is an irrational number. So again "none of the above".

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