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. 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 intersection C). (Enter your answers to three decimal places.) P(A)= 0.44 P(C) = 0.5 P(A intersection 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 a person has type B blood, the probability that he is from ethnic group 3 is given by P(A | C). If we know that the individual came from ethnic group 3, the probability that he has type A is given by P(C I 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 ca me from ethnic group 3, the probability that he has type A 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). (c) If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1? _____

Explanation / Answer

Part (a)

P(A) = Probability that a randomly selected individual has Type A blood group

= proportion of individuals having Type A blood group

= 0.109 + 0.141 + 0.190 = 0.440 ANSWER 1

P(C) = Probability that a randomly selected individual belongs to ethnic group 3

= proportion of individuals belonging to ethnic group 3

= 0.215 + 0.190 + 0.075 + 0.020 = 0.500 ANSWER 2

= Probability that a randomly selected individual has Type A blood group and also belongs to ethnic group 3

= proportion of individuals having Type A blood group and also belonging to ethnic group 3

= 0.190 ANSWER 3

Part (b)

P(A/C) = probability that a randomly selected individual has Type A blood group given that he/she belongs to ethnic group 3

= P(A C)/ P(C)

= 0.190/0.5 [vide Answer 2 and 3 of Part (a)]

= 0.380 ANSWER 1

P(C/A) = probability that a randomly selected individual belongs to ethnic group 3 given that he/she has Type A blood group

= P(A C)/ P(A)

= 0.190/0.44 [vide Answer 1 and 3 of Part (a)]

= 0.432 ANSWER 2

Interpretation

P(A/C) represents the probability that an individual who is known to belong to ethnic group 3 has blood group A. 5th Answer option ANSWER 3

P(C/A) represents the probability that an individual who is known to have blood group A belongs to ethnic group 3. 6th Answer option ANSWER 4

Part (c)

Probability that a selected individual known to have not Type B blood group belongs to ethnic group 1 = P(1/BC) = P(1 BC)/P(BC).

Now, P(BC) = 1 - P(B) = 1 – (0.013 + 0.018 + 0.073) = 1 - 0.104 = 0.896

P(1 BC) = 0.082 + 0.109 + 0.004 = 0.195

Thus, P(1/BC) = 0.195/0.896 = 0.218 ANSWER

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