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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 $P(A \cap C)$ .

(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 $\mathrm{B}$ blood, what is the probability that he or she is from ethnic group 1$?$

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alright for this problem, The probability of a is just going to be the some of the probabilities of having blood type A in any group. So that is going to be 0.106 plus 0.141 plus 0.200 which is equal to 0.4 for seven. Probability of C is similarly going to be the some of the probabilities of being in ethnic Group three with any blood type is going to be a point to 15 plus 150.200 plus 0.65 plus 0.20 is equal Thio 0.5 Next, the probability of a intersected with me is just the going to be the probability, knowing that we are in blood type A that we will have ethnic group three. So the probability of a intersected with B is 0.200 You could get that straight from the table. Next the probability of a given see So we know that we're in ethnic Group three. What's the probability of having blood Type A that is equal to he of a Intersect see divided by probability of C, which is going to give us just But you can plug in the numbers there That gives us 0.4 Similarly similarly, the probability of C given a is probability of a Intersect see divided by the probability of a which in our case, is going to be about. Actually, I'll put all the decimal places that I have 0.4474 to 7 next for the part. See, the way to interpret the question is that we want to find the probability that we are in group one, given that we are not type V. So that is the same as good as one minus the probability of being in group one Given that we are or yeah, given that we are type B, so you need to figure out Well, that's ah probability of Group one given type B that is going to be the same as probability of all right. Oh, yeah. Group one intersect be divided by the probability of being blood type B. Okay, The probability of being group one and blood type B is 0.8 or sorry. 0.8 Up there 008 then the probability of type of being Type B is point 008 plus 80.18 plus 0.0 point six or plus 0.65 rather 0.65 just going to be that comes out to 0.91 2088 So that is the probability that we are in Group One, given that we do not have blood type B.