A multiple choice examination has 5 questions. Each question has three alternative answers of which exctly one is correct. The probability that a student will get 4 or more correct answers just by guessing is

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$\frac{17}{3^{5}}$

$\frac{13}{3^{5}}$

$\frac{11}{3^{5}}$

$\frac{10}{3^{5}}$

answer is C.

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Detailed Solution

$\begin{array}{rcl} n & = & 5 p = \frac{1}{3} q = \frac{2}{3} \end{array}$

P(getting 4 or more correct answers) $= \begin{array}{rcl} P (x \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 4) + P (x & = & 5) \end{array}$

$\begin{array}{rcl} P (x & = & 4) + P (x = 5) \end{array}$

$= \begin{array}{rcl} {}^{5} \end{array} \begin{array}{rcl} C_{4} \end{array} \begin{array}{rcl} (\frac{1}{3}) \end{array} \begin{array}{rcl} ^{4} \end{array} \begin{array}{rcl} \cdot \end{array} \begin{array}{rcl} (\frac{2}{3}) \end{array} {\begin{array}{rcl} + \end{array}}^{5} \begin{array}{rcl} C_{5} \end{array} \begin{array}{rcl} (\frac{1}{3}) \end{array} \begin{array}{rcl} ^{5} \end{array}$