Question

in the united states, 75 percent of adults wear glasses or contact lenses. a random sample of 10 adults in the united states will be selected. which of the following is closest to the probability that fewer than 8 of the selected adults wear glasses or contact lenses? a 0.10 b 0.28 c 0.47 d 0.53 e 0.76

Explanation:

Step1: Use binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 10\), \(p=0.75\), \(C(n,k)=\frac{n!}{k!(n - k)!}\). We want \(P(X\lt8)=1-[P(X = 8)+P(X = 9)+P(X = 10)]\).

Step2: Calculate \(P(X = 8)\)

\(C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1} = 45\), \(P(X = 8)=C(10,8)\times(0.75)^{8}\times(0.25)^{2}=45\times(0.75)^{8}\times(0.25)^{2}\approx45\times0.1001\times0.0625\approx0.2816\).

Step3: Calculate \(P(X = 9)\)

\(C(10,9)=\frac{10!}{9!(10 - 9)!}=10\), \(P(X = 9)=C(10,9)\times(0.75)^{9}\times(0.25)^{1}=10\times(0.75)^{9}\times0.25\approx10\times0.0751\times0.25\approx0.1877\).

Step4: Calculate \(P(X = 10)\)

\(C(10,10)=\frac{10!}{10!(10 - 10)!}=1\), \(P(X = 10)=(0.75)^{10}\approx0.0563\).

Step5: Calculate \(P(X\lt8)\)

\(P(X\lt8)=1-(0.2816 + 0.1877+0.0563)=1 - 0.5256=0.4744\approx0.47\).

Answer:

C. 0.47