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)?

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