Given,

Total number of questions \(\displaystyle={5}\)

Probability of success \(\displaystyle={\frac{{{1}}}{{{4}}}}\) ( since each question has only 1 correct choice out of 4 choices)

We use binomial distribution here.

Step 2

(a) The probability that he will answer all five questions correctly:

\(\displaystyle{P}{\left({X}={5}\right)}={5}{C}_{{{5}}}{\left({\frac{{{1}}}{{{4}}}}\right)}^{{{5}}}{\left({1}-{\frac{{{1}}}{{{4}}}}\right)}^{{{5}-{5}}}\)

\(\displaystyle={0.000977}\)

(b) The probability that he will answer exactly 2 questions correctly:

\(\displaystyle{P}{\left({X}={2}\right)}=^{{{5}}}{C}_{{{2}}}{\left({\frac{{{1}}}{{{4}}}}\right)}^{{{2}}}{\left({1}-{\frac{{{1}}}{{{4}}}}\right)}^{{{5}-{2}}}\)

\(\displaystyle={0.26367}\)

(c) The probability that he will answer at least 2 questions correctly:

\(\displaystyle{P}{\left({X}\geq{2}\right)}={1}-{P}{\left({X}{ < }{2}\right)}\)

\(\displaystyle={1}-{P}{\left({X}={0}\right)}+{P}{\left({X}={1}\right)}\)

\(\displaystyle={1}-{\left[{0.2373}+{0.3955}\right]}\)

\(\displaystyle={1}-{0.6328}\)

\(\displaystyle={0.3672}\)