Question

Imagine a pet shelter that has noticed older animals (age 5+) have lower adoption rates. This shelter (which has only cats and dogs) has 40% cats and 60% dogs. Twenty percent of the cats are older, and 10% of the dogs are older. The adoption percentage for older cats is 5%, and for older dogs is 10%. Cats that are not older are equally likely to be adopted as not adopted, and this is the same for dogs that are not older. For (b)-(f), write the probability statement and then calculate your answer. a) Draw the probability tree and label all events and probabilities in it. b) What is the probability that a randomly chosen animal is not older, a dog, and is adopted? c) What is the probability that a randomly chosen animal is older, a cat, and is not adopted? d) What is the probability that a randomly chosen animal is adopted? e) Suppose a randomly chosen animal is a cat, what is the probability that it is adopted? f) Suppose a randomly chosen animal is adopted, what is the probability that it is a cat?

Answer 1

Answer:

(a) The tree diagram is attached as an image

(b) P(NO ∩ D ∩ NA) = 0.27

(c) P(O ∩ C ∩ A) = 0.004

(d) P(A) = 0.438

(e) P(A|C) = 0.162

(f) P(C|A) = 0.374

Step-by-step explanation:

Let C denote cats and D denote dogs. Then,

P(C) = 40% = 0.4 and

P(D) = 60% = 0.6

Let O denote that an animal is older and NO denote that an animal is not older. Then for Cats,

P(O) = 20% = 0.2 and

P(NO) = 1 - 0.2 = 0.8

Similarly, for Dogs,

P(O) = 10% = 0.1 and

P(NO) = 1 - 0.1 = 0.9

Let A denote that an animal is adopted and NA denote that an animal is not adopted. For older cats,

P(A) = 5% = 0.05 and

P(NA) = 1 - 0.05 = 0.95

Similarly, for older dogs,

P(A) = 10% = 0.1 and

P(NA) = 1 - 0.1 = 0.9

For cats that are not older,

P(A) = P(NA) = 0.5

For dogs that are not older

P(A) = P(NA) = 0.5

(a) We can illustrate this information using a tree diagram as shown in the image attached. The first stage represents whether the animal is a cat or a dog. Then, they are classified as older or not older and then they are classified according to if they are adopted or not.

(b) We need to compute the probability that the selected animal is not older and a dog and is not adopted i.e. P(NO ∩ D ∩ NA). For this we will use the tree diagram and trace out the path D to NO to NA. We need this sample point hence, we will multiply all three probabilities and find out the answer.

P(NO ∩ D ∩ NA) = (0.6)*(0.9)*(0.5)

P(NO ∩ D ∩ NA) = 0.27

(c) Now, we need to compute P(O ∩ C ∩ A). So, using the tree diagram, we will trace out the route from C to O to A and multiply these three probabilities to get our answer.

P(O ∩ C ∩ A) = 0.4 * 0.2 * 0.05

P(O ∩ C ∩ A) = 0.004

(d) To compute the probability that a randomly chosen animal is adopted, we need to see the combinations in our tree diagram where we find adopted (A). The possible combinations can be: (C, O, A), (C, NO, A), (D, O, A) and (D, NO, A). So, the probability of adopted is:

P(A) = (C, O, A) + (C, NO, A) + (D, O, A) + (D, NO, A)

      = 0.4*0.2*0.05 + 0.4*0.8*0.5 + 0.6*0.1*0.1 + 0.6*0.9*0.5

      = 0.002 + 0.16 + 0.006 + 0.27

P(A) = 0.438

(e) Now, we are given that the chosen animal is a cat and we need to compute the probability that it is adopted. So, from the tree diagram, we can see that a cat who is oler can be adopted (C, O, A) and a cat who is not older can also be adopted (C, NO, A). These two sample points make up that probability that a cat is adopted. So,

P(A|C) = (C, O, A) + (C, NO, A)

          = 0.4*0.2*0.05 + 0.4*0.8*0.5

          = 0.002 + 0.16

P(A|C) = 0.162

(f) P(C|A) = P(C∩A)/P(A)

    Now this can hold true for older as well as not older cats. We will consider both and add them.

  P(C|A) = P(C∩A)/P(A) (older cats) + P(C∩A)/P(A) (not older cats)

             = (0.4*0.2*0.05)/0.438 + (0.4*0.8*0.5)/0.438

             = 0.004/0.438 + 0.16/0.438

             = 0.009132 + 0.36529

  P(C|A) = 0.374