Question

we polled theme - park enthusiasts from around the country about their favorite amusement parks and attractions. the results are broken down in the following frequency table by park name and attraction type. assume that each voter got to choose only one favorite park and one favorite type of attraction. rows represent the different amusement parks from around the country. columns represent the different types of theme - park attractions. event a: choosing eight pennants as a favorite park event b: choosing a water slide as a favorite ride use the frequency table to compute the following. probability of choosing eight pennants as a favorite park: p(a)= probability of choosing a water slide as a favorite ride: p(b)= probability of choosing eight pennants as a favorite park and choosing a water slide as a favorite ride: p(a and b)= probability of choosing eight pennants as a favorite park or choosing a water slide as a favorite ride: p(a or b)=

Explanation:

Step1: Recall probability formula

The probability of an event \(E\) is \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). The total number of outcomes here is \(264\) (the grand - total in the table).

Step2: Calculate \(P(A)\)

Event \(A\) is choosing Eight Pennants as a favorite park. The number of people who chose Eight Pennants as a favorite park is \(52\). So \(P(A)=\frac{52}{264}=\frac{13}{66}\).

Step3: Calculate \(P(B)\)

Event \(B\) is choosing a water - slide as a favorite ride. The number of people who chose a water - slide as a favorite ride is \(76\). So \(P(B)=\frac{76}{264}=\frac{19}{66}\).

Step4: Calculate \(P(A\cap B)\)

The number of people who chose Eight Pennants as a favorite park and a water - slide as a favorite ride is \(6\). So \(P(A\cap B)=\frac{6}{264}=\frac{1}{44}\).

Step5: Calculate \(P(A\cup B)\)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Substitute the values: \(P(A\cup B)=\frac{13}{66}+\frac{19}{66}-\frac{1}{44}\). First, find a common denominator, which is \(132\). Then \(\frac{13}{66}+\frac{19}{66}-\frac{1}{44}=\frac{26 + 38-3}{132}=\frac{61}{132}\).

Answer:

\(P(A)=\frac{13}{66}\)
\(P(B)=\frac{19}{66}\)
\(P(A\cap B)=\frac{1}{44}\)
\(P(A\cup B)=\frac{61}{132}\)