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.