Question

In a survey of 35 people, 27 have a dog , 23 have a cat , and five neither a dog nor a cat

How many of the 35 people have both a dog and cat ?

What is the probability that a randomly-selected person has both a dog and a cat?


(Round to 3 decimal places)

Answer 1

To find out how many people have both a dog and a cat, we can use the principle of inclusion-exclusion. Let

�

D be the set of people with a dog,

�

C be the set of people with a cat, and

�

N be the set of people with neither.

The formula for the number of people with both a dog and a cat is given by:

∣

�

∩

�

∣

=

∣

�

∣

+

∣

�

∣

−

∣

�

∪

�

∣

∣D∩C∣=∣D∣+∣C∣−∣D∪C∣

Where:

∣

�

∣

∣D∣ is the number of people with a dog (27),

∣

�

∣

∣C∣ is the number of people with a cat (23),

∣

�

∪

�

∣

∣D∪C∣ is the number of people with either a dog or a cat (35 - 5 = 30).

∣

�

∩

�

∣

=

27

+

23

−

30

=

20

∣D∩C∣=27+23−30=20

So, 20 out of the 35 people have both a dog and a cat.

Now, to find the probability that a randomly-selected person has both a dog and a cat, divide the number of people with both by the total number of people:

�

(

both

)

=

∣

�

∩

�

∣

Total

=

20

35

P(both)=

Total

∣D∩C∣

​

=

35

20

​

�

(

both

)

≈

0.571

P(both)≈0.571

So, the probability that a randomly-selected person has both a dog and a cat is approximately

0.571

0.571 when rounded to three decimal places.

Explanation: