Question

cats and dogs
a survey polls 34 participants, and asks them how many dogs they have. they can respond with 0 dogs, 1 dog, or 2+ dogs. the survey similarly asks them how many cats they have. the results of the 34 responses are shown in the table below. note that many people have both dogs and cats.

	0 dogs	1 dog	2+ dogs	tot
1 cat	5	3	6	14
2+ cats	3	3	2	8
total	9	13	12	34

3 fill in the blank 5 points
how many participants own exactly 1 cat?
how many participants own exactly 1 dog?
how many participants own exactly 1 cat and exactly 1 dog?
according to the union rule, how many participants own exactly 1 cat or exactly 1 dog?
use the union rule idea and your answers above to find the probability that a participant owns exactly 1 cat or exactly 1 dog? (enter as fraction, out of 34)

Explanation:

Step1: Recall the union - rule formula

The formula for the union of two events \(A\) and \(B\) is \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\), where \(n(A)\) is the number of elements in event \(A\), \(n(B)\) is the number of elements in event \(B\), and \(n(A\cap B)\) is the number of elements in the intersection of \(A\) and \(B\).
Let \(A\) be the event that a participant owns exactly 1 cat and \(B\) be the event that a participant owns exactly 1 dog.
We are given \(n(A) = 14\), \(n(B)=13\), and \(n(A\cap B)=3\).

Step2: Calculate \(n(A\cup B)\)

Substitute the values into the formula: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)=14 + 13-3\).
First, \(14+13 = 27\), then \(27-3=21\).

Step3: Calculate the probability

The probability \(P\) that a participant owns exactly 1 cat or exactly 1 dog is given by \(P=\frac{n(A\cup B)}{n(S)}\), where \(n(S) = 34\) (total number of participants).
Since \(n(A\cup B)=21\) and \(n(S)=34\), the probability \(P=\frac{21}{34}\).

Answer:

How many participants own exactly 1 cat? 14
How many participants own exactly 1 dog? 13
How many participants own exactly 1 cat and exactly 1 dog? 3
According to the union - rule, how many participants own exactly 1 cat or exactly 1 dog? 21
Use the union - rule idea and your answers above to find the probability that a participant owns exactly 1 cat or exactly 1 dog? (Enter as fraction, out of 34) \(\frac{21}{34}\)