a) are believing that the earth is warming an...
a) are believing that the earth is warming and being a liberal democrat disjoint? disjoint not disjoint b) what is the probability that a randomly chosen respondent believes the earth is warming or is a liberal democrat? (p(\text{warming or liberal})=) (please round to 4 decimal places, if needed) c) what is the probability that a randomly chosen respondent believes the earth is warming given that they are a liberal democrat?
Answer
# Explanation:
## Step1: Define disjoint events
Two events are disjoint if they cannot occur at the same time. Since there are liberal Democrats who believe the earth is warming (0.2554 in the table), the events are not disjoint.
## Step2: Use the addition - rule for probability
The formula for \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event that a respondent believes the earth is warming and \(B\) be the event that a respondent is a liberal Democrat. \(P(A) = 0.63\), \(P(B)=0.3279\), and \(P(A\cap B)=0.2554\). Then \(P(A\cup B)=0.63 + 0.3279- 0.2554\).
\[P(A\cup B)=0.7025\]
## Step3: Use the formula for conditional probability
The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is the event that a respondent believes the earth is warming and \(B\) is the event that a respondent is a liberal Democrat. \(P(A\cap B) = 0.2554\) and \(P(B)=0.3279\). Then \(P(A|B)=\frac{0.2554}{0.3279}\approx0.7789\)
# Answer:
a) not disjoint
b) 0.7025
c) 0.7789