Questions: 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 O A B AB Ethnic Group 2 0.082 0.112 0.009 0.004 3 0.129 0.141 0.018 0.005 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 ∩ C). (Enter your answers to three decimal places.) P(A) = 0.447 P(C) = 0.5 P(A ∩ C) =

Solution Steps

To solve this problem, we need to calculate the probabilities of specific events based on the given joint probability table.

1. Calculate \( P(A) \): Sum the probabilities of blood type A across all ethnic groups.
2. Calculate \( P(C) \): Sum the probabilities of all blood types within ethnic group 3.
3. Calculate \( P(A \cap C) \): Find the probability of blood type A within ethnic group 3.

To find the probability of selecting an individual with blood type A, we sum the probabilities of blood type A across all ethnic groups: \[ P(A) = P(A \mid \text{ethnic group 2}) + P(A \mid \text{ethnic group 3}) \] Given: \[ P(A \mid \text{ethnic group 2}) = 0.112 \] \[ P(A \mid \text{ethnic group 3}) = 0.141 \] Thus: \[ P(A) = 0.112 + 0.141 = 0.253 \]

To find the probability of selecting an individual from ethnic group 3, we sum the probabilities of all blood types within ethnic group 3: \[ P(C) = P(O \mid \text{ethnic group 3}) + P(A \mid \text{ethnic group 3}) + P(B \mid \text{ethnic group 3}) + P(AB \mid \text{ethnic group 3}) \] Given: \[ P(O \mid \text{ethnic group 3}) = 0.129 \] \[ P(A \mid \text{ethnic group 3}) = 0.141 \] \[ P(B \mid \text{ethnic group 3}) = 0.018 \] \[ P(AB \mid \text{ethnic group 3}) = 0.005 \] Thus: \[ P(C) = 0.129 + 0.141 + 0.018 + 0.005 = 0.293 \]

To find the probability of selecting an individual with blood type A from ethnic group 3, we use the given probability: \[ P(A \cap C) = P(A \mid \text{ethnic group 3}) = 0.141 \]

Final Answer