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A multiple choice examination has $10$ questions, each of which has $4$ possible answers.Suppose that for each question a student knows the correct answer with probability $0.8$ and guesses with probability $0.2$. Find the probability that the student will correctly answer exactly $9$ questions?

My solution:

P(Exactly 9 correct answers)= $10$ * $(0.8)^9$ * $0.2$
The given solution answer is $0.3474$.
How should I solve this?

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    $\begingroup$ If the idea is that the student guesses if he doesn't know the answer, then the chance of answering any question correctly is $P(\text{knows}) + P(\text{guesses correctly}) = 0.8 + 0.2 \times 0.25$. Does that help? $\endgroup$ – Cehhiro Oct 14 '16 at 22:53
  • $\begingroup$ This is much clear. Thanks. $\endgroup$ – roang Oct 14 '16 at 22:59
  • $\begingroup$ @O.VonSeckendorff Why 0.25 and not 0.5 ? For example: $0.8 + 0.2 \times \underbrace{0.5}_{\uparrow} = 0.81$. $\endgroup$ – Felix Marin Oct 15 '16 at 2:31
  • $\begingroup$ Because there are 4 options and each has equal probability. $\endgroup$ – roang Oct 15 '16 at 2:33
  • $\begingroup$ @roang Got it !!!. Thanks. $\endgroup$ – Felix Marin Oct 15 '16 at 2:33
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Basic approach. The student knows the correct answer with probability $0.8$ and guesses with probability $0.2$—presumably, they guess each of the four possible answers with equal likelihood. Therefore, the probability that they submit the correct answer is greater than $0.8$. How much more is it? Use that value instead of $0.8$ in the binomial expansion, and you should obtain the right answer.

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    $\begingroup$ P(Submit right ans) = P(Knows ans)P(Submit right answer|knows ans) + P(does n't know ans)P(Submit right answer|does n't know ans) = 0.8 · 1 + 0.2 · 0.25 = 0.8 + 0.05 = 0.85. Is this correct? $\endgroup$ – roang Oct 14 '16 at 22:53
  • $\begingroup$ @roang: Why don't you try using that value and seeing if you get the right answer? :-) $\endgroup$ – Brian Tung Oct 14 '16 at 22:54
  • $\begingroup$ :) Thanks. That was the right answer. $\endgroup$ – roang Oct 14 '16 at 22:58

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