Probability

[nym0112]

Help! Please!

If x, y, and z are selected independently and and random from the interval [0,1], then the probability that x>=yz is...

A. 3/4
B. 2/3
C. 1/2
D. 1/3
E. 1/4

[kanha81]

Help! Please!

If x, y, and z are selected independently and and random from the interval [0,1], then the probability that x>=yz is...

A. 3/4
B. 2/3
C. 1/2
D. 1/3
E. 1/4

Nice question!

This is how I proceeded; however, would like to know OA and other way of solving the problem.

P(E: x>=yz):
1) x=1, y=0, z=1: x>=yz => 1>=(0)*(1)
2) x=0, y=0, z=0: x>=yz => 0>=0
3) x=0, y=1, z=0: x>=yz => 0>=0
4) x=1, y=1, z=1: x>=yz => 1>=1

P(E: x>=yz) = [spoiler]1/4[/spoiler]
[spoiler][E][/spoiler]

Please post OA.

[nym0112]

So do we assume that x, y, and z cannot be fractions, but only 0 or 1?

[Ian Stewart]

Help! Please!

If x, y, and z are selected independently and and random from the interval [0,1], then the probability that x>=yz is...

A. 3/4
B. 2/3
C. 1/2
D. 1/3
E. 1/4

This question is from a GRE subject test in mathematics, so if you're applying to do a PhD in pure math, you might need to be able to answer it. If you're taking the GMAT, don't worry about it. The GMAT tests only discrete probability -- that is, it only tests probability in situations where there are a finite number of outcomes (flipping a coin a few times, picking marbles from a bag, picking an employee at random from a company, etc). The above question tests continuous probability -- when it mentions choosing x, y and z from the interval [0,1], that means that x, y and z could be any of the infinite number of decimals between 0 and 1 -- so you could never see it on the GMAT test.

Still, because of the answer choices, it's fairly easy to see that A must be correct; if x is not the smallest of the three numbers (i.e. if either x > y or x > z), it's absolutely certain that x > yz (since all the numbers are between 0 and 1). There's a 2/3 chance that x is not the smallest element, so the answer is at least 2/3. Since x > yz can be true even when x is the smallest element, the answer must be greater than 2/3, and A is the only answer that makes sense. Still, I can't see any way to actually prove the answer is exactly 3/4 without using calculus.