This seems to be another easy question but I am a bit confused. I know delta is a proxy for an option finishing ITM. Delta also happens to be N(d1) in the BSM pricing model. N(d1) usually is pretty close to N(d2) but not exact and deviates as time to expiration increases. Some sources say that N(d2), is actually the probability of the option expiring in the money.
However, if you look at the equation for N(d1), below, you'll see that it involves "r" which is the result of risk neutral pricing.
A final source mentions that the above d1 equation, involving "r" is actually not accurate for the probability of an option expiring ITM. In fact, this source claims that "r" should be replaced by mu, or the mean return of the underlying. Also the subsequent + sign should be replaced by a - sign. Basically claims that we should examine probabilities in a risk natural world.
So now I am confused. What am I missing? If I really want to calculate the probability of an option finishing ITM, what equation should I use? Is every source right and there are just small caveats I am missing?