You have a 5-question multiple-choice test. Each question has four choices. You don’t know any of the answers. What is the experimental probability that you will guess exactly three out of five questions correctly?

Each question has four choices, then the probability to guess a correct answer is \( p= \frac{1}{4} \) and the probability to select incorrect choice is  \( q=1-p=1- \frac{1}{4} = \frac{3}{4} \).
You have a 5-question multiple-choice test, then n=5. The probability that you will guess exactly three out of five questions correctly is
\( C_n^kp^nq^{n-k}=C_5^3p^3q^{5-3}= \dfrac{5!}{3!\cdot 2!} \left( \dfrac{1}{4} \right)^3\left( \dfrac{3}{4} \right)^2=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5}{1\cdot 2\cdot 3\cdot 1\cdot 2} \cdot \dfrac{9}{1024} = \)
\( = \dfrac{90}{1024} = \dfrac{45}{512} \).


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