Example:Suppose in Karnataka, 51% of the adults are males. One adult is randomly selected for a survey involving credit card usage.
- Find the prior probability that the selected person is a male.
- It is later learnt 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 available data). Use this additional information to find the probability that the selected subject is a male.
Solution
Let us use the following notation:
M = male M̅ = female (or not male)
C = cigar smoker C̅ = not a cigar smoker
- Before using the information given in part b, we know only that 51% of the adults in Karnataka are males, so the probability of randomly selecting an adult and getting a male is given by P(M) = 0.51.
- Based on the additional given information, we have the following:
P (M) = 0.51 because 51% of the adults are males.
P (M̅) = 0.49 because 49% of the adults are females (not males)
P (C│M) = 0.095 because 9.5% of the males smoke cigars (That is, the probability of getting someone who smokes cigars, given that the person is a male, is 0.095).
P (C│M̅) = 0.017 because 1.7% of the females smoke cigars (That is, the probability of getting someone who smokes cigars, given that the person is a female, is 0.017).
Let us now apply Bayes’ theorem by using the preceding formula with M in place of A, and C in place of B. we get the following result:
P (M). P(C│M)
P (M│C) = ————————————————
[P (M). P (C│M)] + [P (M̅). P (C│M̅)]
0.51* 0.095
= ——————————————–
[0.51* 0.095] + [0.49* 0.017]
= 0.85329341
= 0.853
