This Question is not complete because it lacks the appropriate options
Complete question:
A veterinarian surveys her clients and finds that 32 percent of the households have dogs, 25 percent have cats, and 11 percent have both dogs and cats. Let event C be choosing a client who has cats and let event D be choosing a client who has dogs. Which statements are true? Check all that apply.
1. P(C | D) = 0.78
2. P(D | C) = 0.44
3. P(C ∩ D) = 0.11
4. P(C ∩ D) = P(D ∩ C)
5. P(C | D) = P(D | C)
Answer:
3. P(C ∩ D) = 0.11
4. P(C ∩ D) = P(D ∩ C)
Explanation:
Clients that have Cats = P(C) = 25% = 0.25
Clients that have Dogs = P(D) = 32% = 0.32
The number of Clients that have Dogs and Cats = P(D ∩ C) = 11% = 0.11
Sets of data in mathematics are governed by laws which include Distributive law, Commutative law, Associative law, Identity law e.t.c.
The commutative law of sets states that :
A ∩ B = B ∩ A
A ∪ B = B ∪ A
Therefore, based on the commutative law of sets, Option 4 " P(C ∩ D) = P(D ∩ C) " is correct.
Since we are told in the question above that the number of Clients that have Dogs and Cats = P(D ∩ C) = 11% = 0.11 and based on the commutative law that says
P(C ∩ D) = P(D ∩ C)
Therefore, Option 3 " P(C ∩ D) = 0.11"
is correct as well.