Question

question 8
a true - false test has 10 questions. if lisa answers all questions by guessing, what is the probability of obtaining 8 or 9 correct answers?
$\frac{9}{10}$
$\frac{55}{1024}$
$\frac{7}{128}$
$\frac{4}{5}$

Explanation:

Step1: Calculate the probability of getting 8 correct answers

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n = 10\) (number of trials/questions), \(k\) is the number of successes (correct answers), \(p=\frac{1}{2}\) (probability of getting a question correct by guessing).

For \(k = 8\), \(C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1}=45\)

\(P(X = 8)=C(10,8)\times(\frac{1}{2})^{8}\times(\frac{1}{2})^{2}\)

\(P(X = 8)=45\times(\frac{1}{2})^{10}\)

Step2: Calculate the probability of getting 9 correct answers

For \(k = 9\), \(C(10,9)=\frac{10!}{9!(10 - 9)!}=10\)

\(P(X = 9)=C(10,9)\times(\frac{1}{2})^{9}\times(\frac{1}{2})^{1}\)

\(P(X = 9)=10\times(\frac{1}{2})^{10}\)

Step3: Calculate the probability of getting 8 or 9 correct answers

\(P(X=8\ or\ X = 9)=P(X = 8)+P(X = 9)\)

\(P(X=8\ or\ X = 9)=(45 + 10)\times(\frac{1}{2})^{10}\)

\(P(X=8\ or\ X = 9)=55\times\frac{1}{1024}=\frac{55}{1024}\)

Answer:

\(\frac{55}{1024}\) (the second option)