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??? ?), andP(A ?C) . b. Calculate both P(A|C) and P(C|A) , and explain in context what each of these probabilities represents. c.If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?

Solution : Step 1: Here population of a particular country consists of three ethnic groups. And each one of them belong to one of the four major blood group. And also it is defined as the events A = { type A selected}, B = { type B selected}, and C = { ethnic group 3 selected}. Step 2 : a) we have to find P(A), P(C), and P(A C) here the corresponding joining probability is given as From the probability table it we can find directly that the P(A)= P( type A selected) =0.106+ 0.141+ 0.200= 0.447 P(C)= P(ethnic group 3 is selected)= .215+.200+.065+.020 =0.5 P(AC)=P( individuals from type A or from ethnic group 3 is selected) = P(A)+ P(C)-P(AC) We know that P(AC)= P(individuals having blood group A and belong to ethnic group 3 is selected)= .200 So p(AC)= 0.447+0.5- .200 = 0.747 Step 3: b) P(AC) 1)We know that P (A/C ) = P(C) So P(AC) = 0.200 P(C)= 0.5 P(A/C)= 0.2/0.5 = 0.4 2) Here P(C/A )= P(AC) P(A) (AC) = 0.200 P(A)= 0.447 =. 0.447 P(A/C) represent the probability that select individuals belong to blood group A that already belong to ethnic group 3. P(C/A) represent the probability that select individuals belong to the ethnic group that already belong to blood group A.