Question

a cbs news poll involved a nationwide random sample of 651 adults, asked those affiliation (democrat, republican or none) and their opinion of how the us economy (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 three decimal places)
a. p(republican) =
b. p(worse) =
c. p(worse|republican) =
d. p(republican|worse) =
e. p(republican and worse) =

Explanation:

Step1: Calculate total number of adults

The total number of adults is the sum of all values in the table. $38 + 104+44 + 12+87+137+21+90+118 = 651$.

Step2: Calculate P(Republican)

The number of Republicans is $38 + 104+44=186$. So $P(\text{Republican})=\frac{186}{651}\approx0.286$ (rounded to three - decimal places).

Step3: Calculate P(worse)

The number of people who think the economy is getting worse is $44 + 137+118 = 299$. So $P(\text{worse})=\frac{299}{651}\approx0.460$ (rounded to three - decimal places).

Step4: Calculate P(worse|Republican)

By the formula for conditional probability $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Here, $A$ is the event "worse" and $B$ is the event "Republican". $P(\text{worse}\cap\text{Republican})=\frac{44}{651}$ and $P(\text{Republican})=\frac{186}{651}$. So $P(\text{worse}|\text{Republican})=\frac{44}{186}\approx0.237$ (rounded to three - decimal places).

Step5: Calculate P(Republican and worse)

The number of Republicans who think the economy is getting worse is 44. So $P(\text{Republican and worse})=\frac{44}{651}\approx0.068$ (rounded to three - decimal places).

Answer:

a. $P(\text{Republican})\approx0.286$
b. $P(\text{worse})\approx0.460$
c. $P(\text{worse}|\text{Republican})\approx0.237$
d. $P(\text{Republican}|\text{worse})\approx0.147$
e. $P(\text{Republican and worse})\approx0.068$