How to check a statement about orbital angles of exoplanets?

[Fabioonier]

Hi, everybody. Mi name is Fabio Onier Osorio Pelaez and I'm from Colombia.

I hope to be finishing my Bachelor´s degree in Physics at University of Antioquia by next August. I'm doing my final project on the detection of planets by the Radial Velocity technique and I have a question about an statement of Christophe Lovis and Debra Fischer in a paragraph in the third page or their article (page 29 in the book Exoplanets). I quote the complete paragraph emphasizing the statement in which I am interested:

"The unknown inclination angle $i$ prevents us from measuring the true mass of the companion $m_2$. While this is an important limitation of the RV technique for individual systems, this fact does not have a large impact on statistical studies of exoplanet populations. Because inclination angles are randomly distributed in space, angles close to 90° (edge-on system) are much more frequent than pole-on configurations. Indeed, the distribution function for $i$ is given by $f(i)di=\sin(i)di$. As a consequence, the average value of $\sin(i)$ is equal to $\pi/4$ (0.79). Moreover, the a priori probability that $\sin(i)$ is larger than 0.5 is 87%."

I wondered if you may tell me how did they conclude that. I'll be thankful if you can help me with that information. It will be very useful for my work.

 

[Simon Bridge]

Welcome to PF;
The authors tell you: "Because the inclination angles are randomly distributed..."

 

[Bandersnatch]

To give you another hint, imagine a six-sided die (i.e. a cube) being thrown. Pick anyone pair of opposite sides - these represent the poles of the orbital arrangement. The remaining four sides represent the edges. As you can already see, there's more possibilities of an edge-on result (4 of 6) than of a pole-on result (2 of 6).
Generalise the cube to a sphere and see if you can get the same result the authors got.