Question

1. 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. 0.5
c. 0.3
d. 0.03125

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

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

For a True/False quiz, \(n = 5\) (number of questions), \(k = 5\) (number of correct answers), and \(p=\frac{1}{2}=0.5\) (probability of getting a single question correct by guessing)

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

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

Step4: Substitute the values into the binomial formula

\(P(X = 5)=C(5,5)\times(0.5)^{5}\times(1 - 0.5)^{5 - 5}\)
\(P(X = 5)=1\times(0.5)^{5}\times(0.5)^{0}\)
Since \(a^{0}=1\) for \(a
eq0\), we have \(P(X = 5)=(0.5)^{5}\)
\((0.5)^{5}=\frac{1}{2^{5}}=\frac{1}{32}=0.03125\)

Answer:

d. \(0.03125\)