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A multiple choice test consists of 5 questions, each question having 4 responses. There is only one correct response and the remaining 3 are incorrect responses. If a candidate attempts all the 5 questions then the probability that he answers at least 3 questions incorrectly is
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Verified Answer
The correct answer is:
$\frac{459}{512}$
Given $x=5, P=\frac{1}{4}, q=\frac{3}{4}$
$P$ (at least there incorrect answer)
$$
\begin{aligned}
& ={ }^5 \mathrm{c}_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2+{ }^5 \mathrm{c}_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right)^1+{ }^5 \mathrm{c}_5\left(\frac{3}{4}\right)^5\left(\frac{1}{4}\right) \\
& =\left(\frac{3}{4}\right)^3\left[\frac{5 \times 4}{2} \times \frac{1}{16}+5 \times \frac{3}{16}+1 \times \frac{9}{16}\right] \\
& =\frac{27}{64}\left[\frac{10}{16}+\frac{15}{16}+\frac{9}{16}\right]=\frac{27}{64} \times \frac{34}{16} \\
& =\frac{459}{512}
\end{aligned}
$$
$P$ (at least there incorrect answer)
$$
\begin{aligned}
& ={ }^5 \mathrm{c}_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2+{ }^5 \mathrm{c}_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right)^1+{ }^5 \mathrm{c}_5\left(\frac{3}{4}\right)^5\left(\frac{1}{4}\right) \\
& =\left(\frac{3}{4}\right)^3\left[\frac{5 \times 4}{2} \times \frac{1}{16}+5 \times \frac{3}{16}+1 \times \frac{9}{16}\right] \\
& =\frac{27}{64}\left[\frac{10}{16}+\frac{15}{16}+\frac{9}{16}\right]=\frac{27}{64} \times \frac{34}{16} \\
& =\frac{459}{512}
\end{aligned}
$$
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