Question

a cbs news poll involved 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

if we randomly select one of the adults who participated in this study, compute: (round to four decimal places)
a. p(democrat) =
b. p(same) =
c. p(same|democrat) =
d. p(democrat|same) =
e. p(democrat and same) =

Explanation:

Step1: Calculate total number of adults

The total number of adults in the sample is \(38 + 104+44 + 12+87 + 137+21+90+118=651\).

Step2: Calculate \(P(\text{Democrat})\)

The number of Democrats is \(12 + 87+137 = 236\). So \(P(\text{Democrat})=\frac{236}{651}\approx0.3625\).

Step3: Calculate \(P(\text{same})\)

The number of people who think the economy is the same is \(104 + 87+90=281\). So \(P(\text{same})=\frac{281}{651}\approx0.4316\).

Step4: Calculate \(P(\text{same}|\text{Democrat})\)

By the formula for conditional - probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\), here \(A\) is the event “thinks the economy is the same” and \(B\) is the event “is a Democrat”. \(P(\text{same}\cap\text{Democrat})=\frac{87}{651}\), \(P(\text{Democrat})=\frac{236}{651}\), so \(P(\text{same}|\text{Democrat})=\frac{87}{236}\approx0.3686\).

Step5: Calculate \(P(\text{Democrat}|\text{same})\)

\(P(\text{Democrat}\cap\text{same})=\frac{87}{651}\), \(P(\text{same})=\frac{281}{651}\), so \(P(\text{Democrat}|\text{same})=\frac{87}{281}\approx0.3096\).

Step6: Calculate \(P(\text{Democrat and same})\)

The number of Democrats who think the economy is the same is 87. So \(P(\text{Democrat and same})=\frac{87}{651}\approx0.1336\).

Answer:

a. \(0.3625\)
b. \(0.4316\)
c. \(0.3686\)
d. \(0.3096\)
e. \(0.1336\)