A CBS News poll conducted June 10 and 11, 2006, among a nationwide random sample of 651 adults, asked those adults about their party affiliation (Democrat, Republican or none) and their opinion of how the US economy was changing ("getting better," "getting worse" or "about the same"). The results are shown in the table below.

Question

better same worse
Republican 38 104 44
Democrat 12 87 137
none 21 90 118

Express your answers as a decimal and round to the nearest 0.001 (in other words, type 0.123, not 12.3% or 0.123456).

If we randomly select one of the adults who participated in this study, compute:

P(Democrat) =

P(worse) =

P(worse|Democrat) =

P(Democrat|worse) =

P(Democrat and worse) =

Answer 1

To compute the probabilities requested, we first need to calculate the total number of respondents and then the relevant counts from the table.

Total Respondents: The total number of respondents is the sum of all values in the table:

\[ Total = (38 + 104 + 44) + (12 + 87 + 137) + (21 + 90 + 118) = 651 \]

1. Calculate \( P(Democrat) \): To find \( P(Democrat) \), we need the number of Democrats, which is the sum of all Democrats in the table:

\[ Number \ of \ Democrats = 12 + 87 + 137 = 236 \] \[ P(Democrat) = \frac{Number \ of \ Democrats}{Total} = \frac{236}{651} \approx 0.362 \]

2. Calculate \( P(worse) \): To find \( P(worse) \), we need the total number of people who think the economy is "worse":

\[ Number \ who \ think \ worse = 44 + 137 + 118 = 299 \] \[ P(worse) = \frac{Number \ who \ think \ worse}{Total} = \frac{299}{651} \approx 0.459 \]

3. Calculate \( P(worse|Democrat) \): To find \( P(worse|Democrat) \), we need the number of Democrats who think the economy is "worse":

\[ Number \ of \ Democrats \ who \ think \ worse = 137 \] \[ P(worse|Democrat) = \frac{Number \ of \ Democrats \ who \ think \ worse}{Number \ of \ Democrats} = \frac{137}{236} \approx 0.580 \]

4. Calculate \( P(Democrat|worse) \): To find \( P(Democrat|worse) \), we use the number of Democrats who think the economy is "worse" and the total number who think it is "worse":

\[ P(Democrat|worse) = \frac{Number \ of \ Democrats \ who \ think \ worse}{Number \ who \ think \ worse} = \frac{137}{299} \approx 0.458 \]

5. Calculate \( P(Democrat \ and \ worse) \): To find \( P(Democrat \ and \ worse) \), we need the number of Democrats who think the economy is "worse":

\[ P(Democrat \ and \ worse) = \frac{Number \ of \ Democrats \ who \ think \ worse}{Total} = \frac{137}{651} \approx 0.210 \]

Final Answers:

• \( P(Democrat) \approx 0.362 \)
• \( P(worse) \approx 0.459 \)
• \( P(worse|Democrat) \approx 0.580 \)
• \( P(Democrat|worse) \approx 0.458 \)
• \( P(Democrat \ and \ worse) \approx 0.210 \)