# probability that the student will correctly answer exactly 9 questions

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?

• 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? – Cehhiro Oct 14 '16 at 22:53
• This is much clear. Thanks. – roang Oct 14 '16 at 22:59
• @O.VonSeckendorff Why 0.25 and not 0.5 ? For example: $0.8 + 0.2 \times \underbrace{0.5}_{\uparrow} = 0.81$. – Felix Marin Oct 15 '16 at 2:31
• Because there are 4 options and each has equal probability. – roang Oct 15 '16 at 2:33
• @roang Got it !!!. Thanks. – Felix Marin Oct 15 '16 at 2:33

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.