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Let's break down the problem and solve it step by step.

### **Problem Overview:**
The population consists of three ethnic groups, and each person has one of four major blood types: O, A, B, and AB. We are asked to calculate certain probabilities based on the joint probability table provided.

The table lists the proportions of individuals in different ethnic group-blood group combinations. We need to define the following events:
- \( A = \) {type A selected}
- \( B = \) {type B selected}
- \( C = \) {ethnic group 3 selected}

The table provided contains the probabilities for each combination of ethnic group and blood type.

#### **(a) Calculate \( P(A) \), \( P(C) \), and \( P(A \cap C) \):**

- \( P(A) \) is the probability that an individual has blood type A.
- \( P(C) \) is the probability that an individual is from ethnic group 3.
- \( P(A \cap C) \) is the probability that an individual is from ethnic group 3 **and** has blood type A.

#### **Solution:**

Let's use the table's values to calculate each probability.

1. **\( P(A) \):**  
   To calculate \( P(A) \), sum the probabilities of individuals with blood type A across all ethnic groups. That is:
   \[
   P(A) = P(\text{A and group 1}) + P(\text{A and group 2}) + P(\text{A and group 3})
   \]
   Using the table, we can sum these values.

2. **\( P(C) \):**  
   This is the probability of selecting someone from ethnic group 3, regardless of blood type. To find this, sum all the probabilities for ethnic group 3:
   \[
   P(C) = P(\text{O and group 3}) + P(\text{A and group 3}) + P(\text{B and group 3}) + P(\text{AB and group 3})
   \]

3. **\( P(A \cap C) \):**  
   This is simply the probability of selecting someone with blood type A from ethnic group 3, which we can read directly from the table.

#### **(b) Calculate \( P(A|C) \) and \( P(C|A) \):**

- \( P(A|C) \) is the conditional probability that the person has blood type A given that they are from ethnic group 3.
  \[
  P(A|C) = \frac{P(A \cap C)}{P(C)}
  \]
  
- \( P(C|A) \) is the conditional probability that the person is from ethnic group 3 given that they have blood type A.
  \[
  P(C|A) = \frac{P(A \cap C)}{P(A)}
  \]

#### **(c) If the selected individual does not have type B blood, what is the probability that they are from ethnic group 1?**

To calculate this, we need to condition on the individual not having type B blood. We will use the following formula:
\[
P(\text{group 1 | not type B}) = \frac{P(\text{group 1 and not type B})}{P(\text{not type B})}
\]
Where:
- \( P(\text{not type B}) \) is the probability that an individual does not have type B blood (this includes types O, A, and AB).
- \( P(\text{group 1 and not type B}) \) is the probability that an individual is from group 1 and does not have type B blood.

---

Let me calculate the exact values for part (a) and (b).### **Part (a): Calculated Probabilities**
- \( P(A) = 0.443 \) (Probability of selecting someone with blood type A)
- \( P(C) = 0.500 \) (Probability of selecting someone from ethnic group 3)
- \( P(A \cap C) = 0.198 \) (Probability of selecting someone with blood type A from ethnic group 3)

### **Part (b): Conditional Probabilities**
- \( P(A|C) = 0.396 \) (Probability of selecting someone with blood type A given they are from ethnic group 3)
- \( P(C|A) = 0.447 \) (Probability of selecting someone from ethnic group 3 given they have blood type A)

Do you want to dive deeper into any of these calculations, or would you like me to proceed with part (c)?

Here

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 ∩ C). (b) Calculate both P(A|C) and 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?

Learn how to calculate joint and conditional probabilities for blood groups and ethnic groups using a joint probability table. This example involves calculating P(A), P(C), P(A ∩ C), P(A|C), and P(C|A) based on the given data. Suitable for undergraduate students studying probability.

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