Question

4. [10 pts] Although cats generally possess an acute sense of hearing, due to an anomaly in their genetic makeup, deafness among white cats with blue eyes is quite common. Approximately 95% of the general cat population are non-white cats (i.e., not pure white), and congenital deafness is extremely rare in non-white cats. However, among white cats, approximately 75% with two blue eyes are deaf, 40% with one blue eye are deaf, and only 19% with eyes of other colors are deaf. Additionally, among white cats, approximately 23% have two blue eyes, 4% have one blue eye, and the remainder have eyes of other colors. What is the probability that a randomly chosen white cat is not deaf?

Answer 1

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4.
Let total cats be 10000
non-white cats = 95% or 9500, deafness rare
white cats = 500, out of which:

23% are 2blue eyed of which 75% are deaf
4% are 1blue eyed of which 40% are deaf
73% are other eyed of which 19% are deaf

P(a randomly chosen white cat isn't deaf)
=(prop. who are 2 blue eye white cats*prop not deaf+prop. who are 1 blue eye white cats*prop not deaf
+prop. who are other eye white cats*prop not deaf)

= (.23)*(1-.75) + (.04)*(1-.40)+(.73)*(1-.19)
= 0.6728

So, probability that a randomly chosen white cat is not deaf is 67.28%