Question

according to a recent poll, half of all pet owners would get their next dog or cat from a shelter. consider a random sample of 25 pet owners and define x as the number of pet owners who would acquire their next dog or cat from a shelter. assume that x is a binomial random variable. complete parts a through f below.
a. for this binomial experiment, define a success.
a. a success is a pet owner who did not acquire their dog or cat from a shelter
b. a success is a pet owner who would not acquire their next dog or cat from a shelter
c. a success is a pet owner who acquired their dog or cat from a shelter.
d. a success is a pet owner who would acquire their next dog or cat from a shelter.
b. for this binomial experiment, what is n?
n = 25
c. for this binomial experiment, what is p?
p = 0.5 (round to one decimal place as needed.)
d. find p(x = 11).
p(x = 11) = (round to three decimal places as needed.)

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\), \(n = 25\), \(k = 11\), \(p=0.5\).

Step2: Calculate combination \(C(25,11)\)

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Step3: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Since \(p = 0.5\), \(k = 11\), \(n=25\), then \(p^{k}=(0.5)^{11}\) and \((1 - p)^{n - k}=(0.5)^{14}\). And \((0.5)^{11}\times(0.5)^{14}=(0.5)^{25}\)

Step4: Calculate \(P(X = 11)\)

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Answer:

\(P(x = 11)=0.133\)