for a precise formulation of statement 2, with a proof, see page 2 of these <A HREF="https://www.cs.princeton.edu/courses/archive/fall13/cos521/lecnotes/lec11.pdf">lecture notes:</A>     
$${\rm Prob}\,\left(|\cos\theta|<\sqrt{\frac{\log d}{d}}\right)>1-\frac{1}{d},$$
where $\theta$ is the angle between two randomly chosen unit vectors in $d$ dimensions.    
<sub>notice a typo at the top of the page, corrected at the bottom, the square root should extend over the entire fraction</sub>