Answer:
(a) 0.48, (b) 0.625, (c) 0.1923
Step-by-step explanation:
First define
Probability a person is a democrat: P(D) = 0.4
Probability a person is a republican: P(R) = 0.6
Probability a person support the measure given that the person is a democrat: P(SM | D) = 0.75
Probability a person support the measure given that the person is a republican: P(SM | R) = 0.3
Now for the Theorem of total probabilities we have
(a) P(SM) = P(SM | D)P(D)+P(SM | R)P(R) = (0.75)(0.4)+(0.3)(0.6) = 0.48
and for the Bayes' Formula we have
(b) P(D | SM) = P(SM | D)P(D)/[P(SM | D)P(D)+P(SM | R)P(R)] = (0.75)(0.4)/0.48 = 0.625
Now let SMc be the complement of support the measure, i.e.,
P(SMc | D) = 0.25 : Probability a person does not support the measure given that the person is a democrat
P(SMc | R) = 0.7: Probability a person does not support the measure given that the person is a republican,
and also for the Bayes' Formula we have
(c) P(D | SMc) = P(SMc | D)P(D)/[P(SMc | D)P(D)+P(SMc | R)P(R)] = (0.25)(0.4)/[(0.25)(0.4)+(0.7)(0.6)] = 0.1/(0.52)=0.1923
