Quantitative Analysis for Management, 13/e

Appendix 2.1: Derivation of Bayes’ Theorem

We know that the following formulas are correct:

$$P\left(A|B\right)=\frac{P\left(AB\right)}{P\left(B\right)}$$ (1)

$$P\left(B|A\right)=\frac{P\left(AB\right)}{P\left(A\right)}$$ (2)

$$\left[\text{which can be rewritten as }P\left(AB\right)=P\left(B|A\right)P\left(A\right)\right]\text{ and}$$

$$P\left(B|A\prime \right)=\frac{P\left(A\prime B\right)}{P\left(A\prime \right)}$$

$$\left[\text{which can be rewritten as }P\left(A\prime B\right)=P\left(B|A\prime \right)P\left(A\prime \right)\right].$$ (3)

Furthermore, by definition, we know that

$$\begin{array}{lll}P(B)\hfill & =\hfill & P(AB)+P(A^{\prime }B)\hfill \\ =\hfill & P(B|A)P(A)+P(B|A^{\prime })P(A^{\prime })\hfill \\ \nwarrow \text{from}(2)\text{from}(3)\nearrow \hfill \end{array}$$ (4)

Substituting Equations 2 and 4 into Equation 1, we have

$$\begin{array}{l}\begin{array}{lll}P(A|B)\hfill & =\hfill & \frac{P(AB)}{P(B)}\swarrow \text{from}\text{(2)}\hfill \\ =\hfill & \frac{P(B|A)P(A)}{\underset{}{\underbrace{P(B|A)P(A)+P(B|A^{\prime })P(A^{\prime })}}}\hfill \end{array}\hfill \\ \text{from}\text{(4)}\nearrow \hfill \end{array}$$ (5)

This is the general form of Bayes’ Theorem, shown as Equation 2-5 in this chapter.