The Solution to Normal Probabilities Using RV Sigma ; g(x) , ( Set at 1 Std.) - (To see the starting relationship of equivalent areas between the Normal Curve and RV Sigma)
1) At 1 Std. I obtain "Z" values by taking the Cumulative area the Normal Curve; to 1 Std. = .8431; Additionally, I find the corresponding equivalent area for RV Sigma = .8431;
I then divide the Cumulative area of the Normal by the Cumulative area of the Standard Deviation. ( See the Blue Integrals and the (Pink) " Z " equation Box below )
2) Normal Probabilities: For Normal Probabilities I take the equivalent area of g(x) at 1 Std. = .3413. I then take that area and divide the area by the area of the Standard Deviation which then gives me the Normal Probabilities.
3) See Normal Probabilities Equation ( Blue )

I have used my formula, RV Sigma to solve for the probabilities of the Normal Curve.
If you look you will see that the ratio of area's between the two distributions can give the Z values
while RV Sigma can by itself solve for the Normal Probabilities.