E(yz) = E(y)*E(z) = 1/2 * 1/2 = 1/4
P(x > 1/4) = 3/4
Can anyone explain why the above method works?
If I consider the expectation of X instead,
P(yz > 1/2) != 1/2 ?
Sincere thanks for the clarification.
Since y and z are independent, so the mean of a product is a product of their means. Therefore, E(zy) = E(z)*E(y)=1/2*1/2=1/4.
In order for x > yz => x should be > 1/4. Again, x is chosen randomly(!), so we can figure out this probability easily: (1-1/4)/(1-0)=3/4.
Your method is incorrect, because using this solution you give an answer to another question.