Chapter 5, Problem 116CP

a.

To determine

Explain the reason behind $P\left(A|B\right)=\frac{P\left(A\text{ and }B\right)}{P\left(B\right)}=\frac{\left[P\left(A\right)P\left(B|A\right)\right]}{P\left(B\right)}$ using conditional probability for $P\left(A|B\right)$ and for $P\left(B|A\right)$.

b.

To determine

Explain the reason behind $P\left(B\right)=P\left(B\text{ and }A\right)+P\left(B\text{ and }{A}^c\right)$ by splitting the event that B occurs into two parts according to whether A occurs.

c.

To determine

Explain the reason for $P\left(B\right)=P\left(A\right)P\left(B|A\right)+P\left(A^c\right)P\left(B|A^c\right)$ using part b and the definition of conditional probability.

d.

To determine

Explain the reason for $P\left(A|B\right)=\frac{P\left(A\right)\times P\left(B|A\right)}{P\left(A\right)\times P\left(B|A\right)+P\left(A^c\right)P\left(B|A^c\right)}$ by combining the formulas in part a to c.