Answer:
Option D -0.67
Step-by-step explanation:
Given : Preference of Brand A =30% ⇒ [tex]P(A)=\frac{30}{100}=0.3[/tex]
Preference of Brand B =70% ⇒ [tex]P(B)=\frac{70}{100}=0.7[/tex]
Prefer Brand A and are Female = 20% ⇒ [tex]P(A/F)=\frac{20}{100}=0.2[/tex]
Prefer Brand B and are Female = 40% ⇒ [tex]P(B/F)=\frac{40}{100}=0.4[/tex]
To find : Selected consumer is female, given that the person prefers Brand A [tex]P(F/A)[/tex]
Solution : Using Bayes' theorem, which state that
[tex]P(A/B)=\frac{P(B/A)P(A)}{P(B)}[/tex]
where, P(A) and P(B) are probabilities of observing A and B.
P(B/A)= is a conditional probability where event B occur and A is true
P(A/B)= also a conditional probability where event A occur and B is true.
Now, applying Bayes' theorem,
[tex]P(F/A)=\frac{P(A/F)P(A)}{P(B)P(B/F)+P(A)P(A/F)}[/tex]
[tex]P(F/A)=\frac{(0.2)(0.3)}{(0.7)(0.4)+(0.3)(0.2)}[/tex]
[tex]P(F/A)=\frac{0.6}{0.28+0.6}[/tex]
[tex]P(F/A)=\frac{0.6}{0.88}[/tex]
[tex]P(F/A)=0.68[/tex]
Therefore, Option D is correct probability that a randomly selected consumer is female, given that the person prefers Brand A -0.67
