Bayes' Theorem_Examples - solution (2)

Example) 

In Orange County 51% of adults are males. An adult is randomly selected for a survey concerning credit card usage.

a. What is the prior probability that the selected adult is a male.

b. It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars (based on data from the Substance Abuse and Mental Health Services Administration). Use this additional information to find the probability that the selected subject is a male.

$\triangleright$ Notation First!

M=male $\rightarrow M^c$ = female

C= cigar smoker $\rightarrow C^c$= not a cigar smoker  

$\triangleright$ Solution (a)

Question (a) is asking what the P(M) is  $\rightarrow$ P(M)=0.51

$\triangleright$ Solution (b)

Question (b) is asking what the P(M|C) is. 

P(M)=0.51, P($M^c$)=0.49

P(C|M)= probability of selecting male who smokes cigar = 0.095

P(C|$M^c$ ) = probability of selecting female who smokes cigar=0.017

Now, $P(M|C)= \frac {P(C|M)\cdot P(M)}{P(C|M)\cdot P(M)+P(C|M^c)\cdot P(M^c)}= \frac {0.095 \cdot 0.51}{0.095\cdot 0.51 + 0.017 \cdot 0.49}$  = 0.853