Question

a test consists of 10 true/false questions. to pass the test a student must answer at least 6 questions correctly. if a student guesses on each question, what is the probability that the student will pass the test? round to three decimal places.
a. 0.377
b. 0.828
c. 0.172
d. 0.205

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success on a single - trial, and \(C(n,k)=\frac{n!}{k!(n - k)!}\). Here, \(n = 10\), \(p=0.5\) (probability of guessing a True/False question correctly), and we need to find \(P(X\geq6)=P(X = 6)+P(X = 7)+P(X = 8)+P(X = 9)+P(X = 10)\)

Step2: Calculate \(C(n,k)\) for each \(k\)

• For \(k = 6\):

\(C(10,6)=\frac{10!}{6!(10 - 6)!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210\)
\(P(X = 6)=C(10,6)\times(0.5)^{6}\times(0.5)^{4}=210\times(0.5)^{10}\)

• For \(k = 7\):

\(C(10,7)=\frac{10!}{7!(10 - 7)!}=\frac{10\times9\times8}{3\times2\times1}=120\)
\(P(X = 7)=C(10,7)\times(0.5)^{7}\times(0.5)^{3}=120\times(0.5)^{10}\)

• For \(k = 8\):

\(C(10,8)=\frac{10!}{8!(10 - 8)!}=\frac{10\times9}{2\times1}=45\)
\(P(X = 8)=C(10,8)\times(0.5)^{8}\times(0.5)^{2}=45\times(0.5)^{10}\)

• For \(k = 9\):

\(C(10,9)=\frac{10!}{9!(10 - 9)!}=10\)
\(P(X = 9)=C(10,9)\times(0.5)^{9}\times(0.5)^{1}=10\times(0.5)^{10}\)

• For \(k = 10\):

\(C(10,10)=\frac{10!}{10!(10 - 10)!}=1\)
\(P(X = 10)=C(10,10)\times(0.5)^{10}\times(0.5)^{0}=1\times(0.5)^{10}\)

Step3: Sum up the probabilities

\(P(X\geq6)=(210 + 120+45 + 10+1)\times(0.5)^{10}\)
\(P(X\geq6)=386\times\frac{1}{1024}\approx0.377\)

Answer:

A. 0.377