Question

On a multiple choice test, if you randomly guessed on three questions, then what is the probability you got at least one of them correct?

Answer 1

Answer:

Number of Questions =3

Probability of giving a correct answer

                              $=\frac{1}{3}$

Probability of giving two correct answers

                       $=\frac{2}{3}$

Probability of giving all correct answers

            $=\frac{3}{3}\\\\=1$

Probability that at least one of them is correct

         $=_{1}^{3}\textrm{C}\\\\=\frac{3!}{(3-1)! \times1!}\\\\=3 \text{ways}\\\\=\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \\\\=\frac{1}{27}$

Probability that two of them is correct

         $=_{2}^{3}\textrm{C}\\\\=\frac{3!}{(3-2)! \times2!}\\\\=3 \text{ways}\\\\=\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \\\\=\frac{8}{27}$

Probability that all of them is correct

         $=_{3}^{3}\textrm{C}\\\\=\frac{3!}{(3-3)! \times3!}\\\\=1 \text{way}\\\\=1$

So, Required probability

         $=\frac{1}{27} \times \frac{8}{27} \times 1\\\\=\frac{8}{729}$