Question

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?

Answer 1

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

$C_n^kp^nq^{n-k}=C_5^3p^3q^{5-3}= \dfrac{5!}{3!\cdot 2!} \left( \dfrac{1}{4} \right)^3\left( \dfrac{3}{4} \right)^2=\dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5}{1\cdot 2\cdot 3\cdot 1\cdot 2} \cdot \dfrac{9}{1024} =$
$= \dfrac{90}{1024} = \dfrac{45}{512}$.

Answer 2

Answer with explanation:

Number of Multiple Choice Question=5

Number of choices =4

Out of 4 choices one is correct Option.

Probability of Correct answer P(C→Correct)

  $=\frac{\text{Total favorable Outcome}}{\text{Total Possible Outcome}}=\frac{1}{4}$

Probability of Incorrect answer P(IC→Incorrect Answer)

   $1-\frac{1}{4}=\frac{3}{4}$

→Experimental Probability of guessing three out of five questions correctly

= 3 Correct +2 Incorrect

$=_{3}^{5}\textrm{C}[P(C)]^3\times [P(I C)]^2\\\\=\frac{5!}{3!\times 2!}\times[\frac{1}{4}]^3 \times[\frac{3}{4}]^2\\\\=10 \times \frac{1}{64} \times \frac{9}{16}\\\\=\frac{90}{1024}\\\\=0.87890$

  =0.88(approx)