Badu-took-a-test-consisting-of-3-multiple-choice-questions-with-4-answer-choices-and-5-true-false-type-questions-If-Badu-answers-all-the-questions-by-guessing-randomly-the-chances-of-him-answering

Question Number 134472 by EDWIN88 last updated on 04/Mar/21

$$$$Badu took a test consisting of 3 multiple choice questions with 4 answer choices and 5 true-false type questions. If Badu answers all the questions by guessing randomly, the chances of him answering correctly are only 2 questions

Commented by EDWIN88 last updated on 04/Mar/21

$$\mathrm{Answer}\mathrm{available}$$$$\left(\mathrm{a}\right)0.5\left(\mathrm{b}\right)0.4\left(\mathrm{c}\right)0.3\left(\mathrm{d}\right)0.2\left(\mathrm{e}\right)0.1$$

Commented by benjo_mathlover last updated on 04/Mar/21

$$\mathrm{case}\left(1\right)$$$$\mathrm{P}\left({\mathrm{A}}_1\right)=\left(\begin{array}{c}3\\ 2\end{array}\right){\left(\frac{1}{4}\right)}^2{\left(\frac{3}{4}\right)}^1\times \left(\begin{array}{c}5\\ 0\end{array}\right){\left(\frac{1}{2}\right)}^5{\left(\frac{1}{2}\right)}^0$$$$=\frac{9}{2^6}\times \frac{1}{2^5}=\frac{9}{2^{11}}$$$$\mathrm{case}\left(2\right)$$$$\mathrm{P}\left({\mathrm{A}}_2\right)=\left(\begin{array}{c}3\\ 1\end{array}\right){\left(\frac{1}{4}\right)}^1{\left(\frac{3}{4}\right)}^2\times \left(\begin{array}{c}5\\ 1\end{array}\right){\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^4$$$$=\frac{27}{4^3}\times \frac{5}{2^5}=\frac{135}{2^{11}}$$$$\mathrm{case}\left(3\right)$$$$\mathrm{P}\left({\mathrm{A}}_3\right)=\left(\begin{array}{c}3\\ 0\end{array}\right){\left(\frac{1}{4}\right)}^0{\left(\frac{3}{4}\right)}^3\times \left(\begin{array}{c}5\\ 2\end{array}\right){\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^3$$$$=\frac{27}{2^6}\times \frac{10}{2^5}=\frac{270}{2^{11}}$$$$\mathrm{totally}\Rightarrow \mathrm{P}\left(\mathrm{A}\right)=\frac{9+135+270}{2^{11}}\approx 0.202148$$$$$$

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