The population of a particular country consists of three ethnic groups. Each ind

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 AB 1 0.082 0.109 0.008 0.004 Ethnic Group 2 0.132 0.141 0.018 0.006 3 0.215 0.191 0.074 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 n C). (Enter your answers to three decimal places.) P(A) P(C) P(A n C)- (b) Calculate both P(A I C) and P(C | A). (Round your answers to three decimal places.) 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 A blood, the probability that he is from ethnic group 3 is given by P(CIA) 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(A I C) If a person has type B blood, the probability that he is from ethnic group 3 is given by P(CIA) If we know that the individual came from ethnic group 3, the probability that he has type A is given by P(AC) (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.)

Explanation / Answer

a) P(A) =P(type A is selected) =0.109 +0.141 +0.191 =0.441

P(B) =P(type B is selected) = 0.008+0.018 +0.074=0.1

P(C) =P(ethnic group 3 selected) = 0.215 +0.191+0.074 +0.020= 0.5

P(A,C) =P(type A & ethnic group 3 selected) =0.191

b) P(A|C)=P(A,C)/P(C)=P(type A ethnic group 3 selected)/P(ethnic group 3 selected) = 0.191/0.5=0.382

P(C|A)=P(A,C)/P(A)=P(type A ethnic group 3 selected)/P(type A selected) = 0.191/0.441 =0.433

The options which apply are as follows-

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

vi) If a person belongs to ethnic group 3,probabilty of having type A blood group is P(A|C).

c) P(ethnic group 1 given doesnt have type B blood group)=P( ethnic group 1| B complement)

= P( ethnic group 1 ,belongs to type O,A or AB)/P(belongs to O,A or AB)

= P( ethnic group 1 ,belongs to type O,A or AB)/(1-P(B) )

=(0.082 +0.109 +0.004) /(1-0.1) = 0.195/0.9 =0.2167= 0.217

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