When thinking about outcome probability <math>p</math> and the changed outcome probability <math>p\prime</math> by adding <math>1</math> to explanatory variable <math>X_1</math>, the following two equations are obtained,
Because <math>\dfrac{p\prime}{1-p\prime}</math> and <math>\dfrac{p}{1-p}</math> are odds of <math>p\prime</math> and <math>p</math>, respectively,
<math>\frac {\left ( \dfrac{p\prime}{1-p\prime} \right )}{ \left ( \dfrac{p}{1-p} \right ) }</math> is the '''odds ratio''' of [the probability when <math>1</math> is added to <math>X_1</math>] to [the probability before adding].
Thus, converted <math>b_1</math> to <math>\color{red}{e^{b_1}}</math> or <math>\color{red}{\exp (b_1)}</math> gives the '''odds ratio of outocome probabilities''' before and after variable <math>X_1</math> gains <math>1</math>.