Question

according to government data, the probability that an adult was never in a museum is 15%. in a random survey of 10 adults, what is the probability that at least eight were in a museum? round to three decimal places
a. 0.800
b. 0.820
c. 0.002
d. 0.200

Explanation:

Step1: Calculate the probability of an adult being in a museum

The probability that an adult was never in a museum is \(p_{never}=0.15\). So the probability that an adult was in a museum is \(p = 1 - 0.15=0.85\). The number of trials \(n = 10\).

Step2: Use the binomial probability formula

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

We want to find \(P(X\geq8)=P(X = 8)+P(X = 9)+P(X = 10)\)

For \(k = 8\):
\(C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1} = 45\)
\(P(X = 8)=C(10,8)\times(0.85)^{8}\times(0.15)^{2}\)
\(P(X = 8)=45\times(0.85)^{8}\times(0.15)^{2}\approx45\times0.272490524\times0.0225\approx0.2759\)

For \(k = 9\):
\(C(10,9)=\frac{10!}{9!(10 - 9)!}=10\)
\(P(X = 9)=C(10,9)\times(0.85)^{9}\times(0.15)^{1}\)
\(P(X = 9)=10\times0.231616945\times0.15\approx0.3474\)

For \(k = 10\):
\(C(10,10)=\frac{10!}{10!(10 - 10)!}=1\)
\(P(X = 10)=(0.85)^{10}\approx0.1969\)

Step3: Sum up the probabilities

\(P(X\geq8)=P(X = 8)+P(X = 9)+P(X = 10)\)
\(P(X\geq8)\approx0.2759+0.3474 + 0.1969=0.8202\approx0.820\)

Answer:

B. 0.820