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We are given a joint probability distribution table for a state congress categorized by legislative group and political party. The goal is to determine the following probabilities:

- \( P(P_1) \): The probability of selecting a Democratic member ( \( P_1 \) )
- \( P(C_1) \): The probability of selecting a Representative ( \( C_1 \) )
- \( P(P_1 \cap C_1) \): The joint probability of selecting a Democratic Representative (both events \( P_1 \) and \( C_1 \))

Let's solve them step by step:

### 1. Probability of selecting a Democratic member \( P(P_1) \)
From the table, we can directly read off \( P(P_1) \) as 0.486. This represents the marginal probability that a randomly chosen member is Democratic.

\[
P(P_1) = 0.486
\]

### 2. Probability of selecting a Representative \( P(C_1) \)
The probability \( P(C_1) \) is the sum of all the probabilities in the "Rep" column (the marginal probability for Representatives).

From the table:
\[
P(C_1) = 0.372 + 0.413 + 0.002 = 0.787
\]

### 3. Joint probability of selecting a Democratic Representative \( P(P_1 \cap C_1) \)
The joint probability of selecting a Democratic Representative is given directly in the table at the intersection of \( P_1 \) and \( C_1 \), which is 0.372.

\[
P(P_1 \cap C_1) = 0.372
\]

### Final Answers:
- \( P(P_1) = 0.486 \)
- \( P(C_1) = 0.787 \)
- \( P(P_1 \cap C_1) = 0.372 \)

Would you like further clarification on these answers, or help with additional questions?

### Related Questions:
1. What is the probability of selecting a Republican Senator?
2. What is the probability of selecting a member from the "Other" category?
3. Calculate \( P(C_2 \mid P_1) \), the conditional probability of being a Senator given that the member is Democratic.
4. What is the marginal probability of selecting a Senator \( P(C_2) \)?
5. What is the conditional probability \( P(P_2 \mid C_1) \), the probability of selecting a Republican given that they are a Representative?

### Tip:
When analyzing joint probability tables, remember that the marginal probabilities are the sums of the individual rows or columns, while the joint probabilities are at the intersections.

Below is a joint probability distribution for the members of a state congress by legislative group and political party. Determine P(P1), P(C1), and P(P1 & C1).

Learn how to determine the joint and marginal probabilities from a probability distribution table involving a state congress categorized by political party and legislative group. This problem includes calculating P(P1), P(C1), and the joint probability P(P1 & C1). Suitable for students in Grades 10-12.

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