Example 5. Select a person at random from a population of 3 ethnic groups A, B,

Question

Example 5. Select a person at random from a population of 3 ethnic groups A, B, and C. Suppose that group A represents 50% of the population while B, and C represent 30% and 20% of the population, respectively. It is known that 1% of A, 0.5% of B and 2% of C will test positive for Tuberculosis. (i) What is the probability that the person will test positive for Tuberculosis? (ii) It is know that the person tested positive for Tuberculosis, What is the proba- bility that he or she was from group C?

Explanation / Answer

(i)

                        Group A             Group B                 Group C                  Total

Test positive 0.50X0.01=0.005    0.30X0.005=0.0015   0.20X0.02=0.004    0.0105

Test negative   0.495                  0.2985                     0.196                    0.9895

Total               0.50                      0.30                         0.20                      1.00

P(Test positive) = 0.0105

(b)

P(Group C/Test positive) = P(Group C & Test positive)/P(Test positive) = 0.004/0.0105 = 0.3810

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