3) An animal shelter has 50 dogs and 30 quarantined. Among the dogs there are 8

Question

3) An animal shelter has 50 dogs and 30 quarantined. Among the dogs there are 8 that are quaranti cats. Among the cats there are 10 that are ned. a) If one of these animals is randomly selected, find the probability that it is quarantined, given that it is a dog. Give the exact answer as a fraction. b) If one of these animals is randomly selected, find the probability that it is a dog, given that it is quarantined. Give the exact answer as a fraction. c) If one of these animals is randomly selected, find the probability that it is a cat, given that it is quarantined. Give the exact answer as a fraction. d) If one of these animals is randomly selected, find the probability that it is quarantined, given that it is a cat. Give the exact answer as a fraction. 30 e) If 3 of these animals are randomly selected, find the probability that at least one isa dog. Give the exact answer as a fraction& the decimal answer rounded to eight decimal places. Show the work. Exact answer: Decimal approx:

Explanation / Answer

here total 80 animals ( 50 dogs + 30 cats)

P(dog)=50/80=5/8 and P(cat)=30/80=3/8

total 18 ( 8 dog + 10 cats) are quarantined

P(quarantined)=18/80=9/40

(a)P(quarantined | dog)=P(quarantined and dog)/P(dog)=(8/80)/(50/80)=8/50=4/25

(b) P(dog | quarantined)=P(quarantined and dog)/P(quarantined)=(8/80)/(18/80)=8/18=4/9

(c)P(cat | quarantined)=P(quarantined and cat)/P(quarantined)=(10/80)/(18/80)=10/18=5/9

(d)P(quarantined | cat)=P(quarantined and cat)/P(cat)=(10/80)/(30/80)=10/30=1/3

(e) let p=P(dog)=5/8

let x=number of dog in selected 3 animal

P(X=at least one dog)=P(X=1)+P(X=2)+P(X=3)=1-P(X=0)=1-(1-5/8)3=1-(3/8)*(3/8)*(3/8)=0.94726563

or we can use binomial distribution with parameter n=3 and p=5/8 and for

Binomial distribution ,P(X=r)=nCrpr(1-p)n-r  

P(X is at least 1)=P(X>=1)=1-P(X=0)=1-0.5273438=0.94726563 (using ms-excel=BINOMDIST(0,3,5/8,1)

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