Maria and John have decided that once they live in a house, they want to have a pet. They go to an animal shelter and find several pets that they would love to take home. There are 30 cats, 4 German Shepherds, 10 Labrador Retrievers, and 22 mixed-breed dogs. Since they can't decide, they place all the adoption cards in a container and draw one. Answer each of the following questions separately, showing all your work to reach each answer.

A. What is the probability that they select a cat?

B. What is the probability that they select either a German Shepherd or a Labrador Retriever? C. What is the probability that if they select a dog, that it is not a mixed breed?

D. If they decide to purposely choose 2 of the 36 available dogs rather than randomly choosing any 2, how many combinations of 2 dogs are possible?

Step-by-step explanation

A).

 

There are 30 cats, 4 German Shepherds, 10 Labrador Retrievers, and 22 mixed-breed dogs in animal shelter.

Total number of pets available for adaption = 30 + 4 + 10 + 22 = 66

Total number of possible ways of selecting any one pet from 66 pets is n(S) = 66C₁ = 66

Total number of cats available for adaption = 30

Number of possible ways of picking a cat from 30 available cats n(E) = 30C₁

Probability of picking a cat, p(E)= no.of ways of selecting a cat / no.of ways of selecting any one of pet

p(E) = ?n(S)n(E)??

p(E) = ?6630??

p(E) = 0.4545

 

B).

 

There are 30 cats, 4 German Shepherds, 10 Labrador Retrievers, and 22 mixed-breed dogs in animal shelter.

Total number of pets available for adaption = 30 + 4 + 10 + 22 = 66

Total number of possible ways of selecting any one pet from 66 pets is n(S) = 66C₁ = 66

Number of german shepherds and Labrador retrievers available together = 4 + 10 = 14

Number of possible ways of selecting either a German Shepherd or a Labrador Retriever, n(E) = 14C₁ = 14

Probability of picking a german shepherd or Labrador retriever, p(E)= no.of ways of selecting either a German Shepherd or a Labrador retriever / no.of ways of selecting any one of pet

p(E) = ?n(S)n(E)??

p(E) = ?6614??

p(E) = 0.2121

 

C).

 

The dog pets available are 4 German Shepherds, 10 Labrador Retrievers, and 22 mixed-breed dogs.

Total number of dog pets = 4 + 10 + 22 = 36

Total number ways of selecting a dog from all dogs n(S) = 36C₁ = 36

 

Number of dogs that are not mixed breeds = 4 + 10 = 14

Number of ways of selecting a dog that is not mixed n(E) = 14C₁ = 14

 

The probability of selecting a non mixed breed from available dogs p(E) = number of ways of selecting a non mixed breed/number of ways of selecting a dog

p(E) = n(E)/n(S)

p(E) = 14/36

p(E) = 0.3888

 

 

 

Or else another approach can be as follows,

 

 

 

Probability if the event B occurs then event A to occur is,

p(A | B) = ?p(B)p(A∩B)??

 

Total number of pets available for adaption = 30 + 4 + 10 + 22 = 66

Total number of possible ways of selecting any one pet from 66 pets is n(S) = 66C₁ = 66

The dog pets available are 4 German Shepherds, 10 Labrador Retrievers, and 22 mixed-breed dogs.

Total number of dog pets B = 4 + 10 + 22 = 36

Total number ways of selecting a dog from all pets n(B) = 36C₁ = 36

Probability of selecting a dog p(B) = n(B)/n(S) = 36/66

Number of dogs that are not mixed breeds A?∩? B = 4 + 10 = 14

Number of ways of selecting a dog that is not mixed n(A?∩? B) = 14C₁ = 14

Probability of selecting a dog that is not mixed breed p(A?∩? B) = n(A?∩? B)/n(S) = 14/66

Probability if they select a dog then it is not a mixed breed is p(A|B)

p(A|B) = p(A?∩? B)/p(B)

p(A|B) = 14/36

p(A|B) = 0.3888

 

 

D).

 

The number of dogs available is 36

The number of possible ways of selecting any two dogs from available dogs = 36C₂ = ?1∗236∗35?? = 630

There are 630 possible combinations of two dogs from 36 dogs.