I Why Does Equation (5) in Bell's Paper Use < σ⋅a > = 1 - 2θ'/π?

[quiteconfused]

TL;DR Summary

Specifically, the demonstration of a single-particle hidden variable

Hello! I am trying to understand Bell's Inequality and although I can follow the arguments of the inequality as are mentioned in modern texts, something was always bothering me. In section III (Illustration) of the original paper, equation (5) states:

< σ⋅a > = 1 - 2θ'/π​

But for the life of me, I don't know why, and I am not sure how to assemble this by hand, it seems like everywhere I find similar values there's just a "well, obviously this gives: " followed by the above. Trying to put together the integral has me wondering if B(b,λ) is just to be taken as 1? Maybe it's been too long since I did any calculus...

 

[wle]

You don't need to do any calculus. You're averaging $\text{sign} \, \boldsymbol{\lambda} \cdot \boldsymbol{a}'$ over vectors $\boldsymbol{\lambda}$ that satisfy $\boldsymbol{\lambda} \cdot \boldsymbol{p} > 0$ for some given vectors $\boldsymbol{a}'$ and $\boldsymbol{p}$. $\text{sign} \, \boldsymbol{\lambda} \cdot \boldsymbol{a}'$ is $1$ if $\boldsymbol{\lambda} \cdot \boldsymbol{a}' > 0$ and $-1$ if $\boldsymbol{\lambda} \cdot \boldsymbol{a}' < 0$. So the question is: out of all $\boldsymbol{\lambda}$s in the hemisphere $\boldsymbol{\lambda} \cdot \boldsymbol{p} > 0$ what fraction are also in the hemisphere $\boldsymbol{\lambda} \cdot \boldsymbol{a}' > 0$ and what fraction are in the hemisphere $\boldsymbol{\lambda} \cdot \boldsymbol{a}' < 0$?

 

[quiteconfused]

Of course, within minutes of explaining the problem in this post I managed to figure it out randomly while traveling to the pub, imagine my surprise when I come back here to find that you told me the same thing. Thank you for confirming what I suspected - I appreciate you helping me do a sanity check!

Edit: I suppose I should find a way to change my username to "considerablylessconfused"