Question

a) are believing that the earth is warming and being a liberal democrat mutually exclusive? not mutually exclusive mutually exclusive b) what is the probability that a randomly chosen respondent believes the earth is warming or is a liberal democrat? 0.7600 (please round to four decimal places) c) what is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal democrat? 0.5294 (please round to four decimal places) d) what is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative republican? 0.3333 (please round to four decimal places) e) does it appear that whether or not a respondent believes the earth is warming is independent of their party ideology? better in global warming and party ideology are dependent better in global warming and party ideology are independent

Explanation:

Step1: Recall probability - definition

Let \(A\) be the event of believing in global - warming and \(B\) be the event of being a certain party member. Conditional probability is defined as \(P(A|B)=\frac{P(A\cap B)}{P(B)}\).

Step2: Analyze part (a)

Two events are mutually exclusive if \(P(A\cap B) = 0\). Since it is possible for a person to be a liberal Democrat and believe that the earth is warming, they are not mutually exclusive.

Step3: Analyze part (b)

We are given \(P(A\cap B)\) (probability of being a liberal and believing in warming) and \(P(B)\) (probability of being a liberal). Using the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), assume \(P(A\cap B) = 0.34\) and \(P(B)=0.45\) (values are assumed for illustration as full data is not provided in a clean - cut way in the image. But the formula is \(P(\text{believing in warming}|\text{liberal})=\frac{\text{Number of liberal and believing in warming}}{\text{Number of liberals}}\)). If we assume from the table - like structure in the image that the relevant values lead to the correct calculation, we calculate the conditional probability.

Step4: Analyze part (c)

Similar to part (b), for the case of liberal Democrats. Use the conditional - probability formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), where \(A\) is believing in warming and \(B\) is being a liberal Democrat.

Step5: Analyze part (d)

For conservative Republicans, use the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) to find the probability of believing in warming given that the person is a conservative Republican.

Step6: Analyze part (e)

Two events (belief in global warming and party ideology) are independent if \(P(A|B)=P(A)\). We need to check if the probability of believing in global warming is the same regardless of party ideology. We can check this by comparing conditional probabilities \(P(\text{warming}|\text{party})\) with the overall probability of believing in warming \(P(\text{warming})\).

Answer:

a. not mutually exclusive
b. (Value depends on correct data extraction from the image, assume calculation using \(P(A|B)=\frac{P(A\cap B)}{P(B)}\))
c. (Value depends on correct data extraction from the image, assume calculation using \(P(A|B)=\frac{P(A\cap B)}{P(B)}\))
d. (Value depends on correct data extraction from the image, assume calculation using \(P(A|B)=\frac{P(A\cap B)}{P(B)}\))
e. (Check if \(P(\text{warming}|\text{party}) = P(\text{warming})\) for all parties to determine dependence/independence)