Question

The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group-blood group combinations.

		Blood Group
		O	A	B	AB
	1	0.082	0.112	0.005	0.004
Ethnic Group	2	0.129	0.141	0.018	0.009
	3	0.215	0.199	0.066	0.020

Suppose that an individual is randomly selected from the population, and define events by A = {type A selected}, B = {type B selected}, and C = {ethnic group 3 selected}. (a) Calculate P(A), P(C), and P(A ∩ C). (Enter your answers to three decimal places.)

P(A)	=
P(C)	=
P(A ∩ C)	=

(b) Calculate both P(A | C) and P(C | A). (Round your answers to three decimal places.)

P(A | C)	=
P(C | A)	=

Explain in context what each of these probabilities represents. (Select all that apply.)

If we know that the individual came from ethnic group 3, the probability that he has type A is given by P(C | A).

If a person has type B blood, the probability that he is from ethnic group 3 is given by P(A | C).

If a person has type A blood, the probability that he is from ethnic group 3 is given by P(C | A). If a person has type A blood, the probability that he is from ethnic group 3 is given by P(A | C).

If a person has type B blood, the probability that he is from ethnic group 3 is given by P(C | A). If we know that the individual came from ethnic group 3, the probability that he has type A is given by P(A | C).

(c) If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1? (Round your answer to three decimal places.)

Answer 1

		O	A	B	AB	total
	1	0.082	0.112	0.005	0.004	0.203
Ethnic Group	2	0.129	0.141	0.018	0.009	0.297
	3	0.215	0.199	0.066	0.02	0.5
	total	0.426	0.452	0.089	0.033	1

a) P(A)=0.452

P(C)=0.5

P(A n C)=0.199

b)

P(A|C)=0.199/0.5=0.398

P(C|A)=0.199/0.452=0.440

If a person has type A blood, the probability that he is from ethnic group 3 is given by P(C | A)

If we know that the individual came from ethnic group 3, the probability that he has type A is given by P(A | C).
c)

P(1|not B)=(0.082+0.112+0.004)/(1-0.089)=0.217