Bayes' Theorem_Examples - solution (1)

Example) Probability and Random Processes, Oxford, 3ED, p.11 

Only two factories manufacture zoggles. 20% of the zoggles from factory A and 5% from factory B are defective. Factory A produces twice as many zoggles as factory B each week. 1) What is the probability that a zoggles, randomly chosen from a weeks production, is satisfactory? 2) If the chosen zoggle is defective, what is the probability that it came from factory A. 

$\triangleright$ Think First

Let D be the event that the chosen zoggle is defective. 

$\rightarrow$ $D^c$ will be the event that the chosen zoggle is NOT defective.  

Let A be the event it was made from factory A. 

$\rightarrow$ $A^c$ will be the event that was made from factory B.

$\triangleright$ Solution

The question 1) is asking what the P($D^c$) is.

Method 1) 
P($D^c$) = 1 - P(D) =1 - [P(D|A) x P(A) + P(D|P($A^c$) x P(P($A^c$) = $1-(0.2 \cdot \frac{2}{3} + 0.05 \cdot \frac {1}{3})=\frac{51}{60}$ 

Method 2) 
P($D^c$) =P($D^c$|A) x P(A) + P($D^c$|P($A^c$) x P(P($A^c$) = $0.8 \cdot \frac {2}{3}+ 0.95\cdot \frac{1}{3}=\frac{51}{60}$ = 0.85 


The question 2) is asking what the P(A|D) is.

$P(A|D)= \frac {P(A \cap D)}{P(D)}=\frac{P(D|A)\cdot P(A)}{P(D)}= \large{\frac {\frac{1}{5}\cdot \frac{2}{3}}{1-\frac{51}{60}}}$ = 0.8889