Question

if a student guesses all answers on a 5 - question true/false quiz, what is the binomial probability of getting all questions correct?
a. 0.2
b. 0.03125
c. 0.3
d. 0.5

Explanation:

Step1: Identify 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 \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Determine values of \(n\), \(k\), and \(p\)

Here, \(n = 5\) (number of questions), \(k=5\) (number of correct answers), \(p = 0.5\) (probability of getting a True - False question correct by guessing)

Step3: Calculate \(C(n,k)\)

\(C(5,5)=\frac{5!}{5!(5 - 5)!}=\frac{5!}{5!0!}=1\) (since \(0!=1\))

Step4: Calculate \(p^{k}\times(1 - p)^{n - k}\)

\(p^{k}\times(1 - p)^{n - k}=(0.5)^{5}\times(1 - 0.5)^{5 - 5}=(0.5)^{5}\times(0.5)^{0}=(0.5)^{5}\)

Step5: Calculate the final probability

\(P(X = 5)=C(5,5)\times(0.5)^{5}=1\times\frac{1}{32}=0.03125\)

Answer:

B. \(0.03125\)