Question

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.

	better	same	worse
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(better) =

p(better|democrat) =

p(democrat|better) =

p(democrat and better) =

Explanation:

Step1: Calculate total number of adults

The total number of adults in the sample is 651.

Step2: Calculate P(Democrat)

The number of Democrats is \(12 + 87+137=236\). So \(P(\text{Democrat})=\frac{236}{651}\approx 0.363\).

Step3: Calculate P(better)

The number of people who think the economy is getting better is \(38 + 12+21 = 71\). So \(P(\text{better})=\frac{71}{651}\approx0.109\).

Step4: Calculate P(better|Democrat)

By the formula for conditional - probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), and \(P(\text{better}|\text{Democrat})=\frac{\text{Number of Democrats who think better}}{\text{Number of Democrats}}=\frac{12}{236}\approx0.051\).

Step5: Calculate P(Democrat|better)

By Bayes' theorem \(P(\text{Democrat}|\text{better})=\frac{P(\text{better}|\text{Democrat})P(\text{Democrat})}{P(\text{better})}\), also \(P(\text{Democrat}|\text{better})=\frac{12}{71}\approx0.169\).

Step6: Calculate P(Democrat and better)

The number of Democrats who think the economy is getting better is 12. So \(P(\text{Democrat and better})=\frac{12}{651}\approx0.018\).

Answer:

P(Democrat) = 0.363
P(better) = 0.109
P(better|Democrat) = 0.051
P(Democrat|better) = 0.169
P(Democrat and better) = 0.018