Question

ramone has 5 difficult questions left to answer on a multiple choice test. Each question has 3 choices. For the first 2 of these questions, he eliminated 1 of the 3 choices. find the probability that he will answer the first 2 questions, as well as at least 2 of the 3 remaining questions correctly.

Answer 1

1. a b c
2. a b c
3. a b c
4. a b c
5. a b c

then she eliminated 1 choice in 1 and 2, say as follows

1.    b c
2. a b 
3. a b c
4. a b c
5. a b c

Probability of answering correctly the first 2, and at least 2 or the remaining 3 is 
P(answering 1,2 and exactly 2 of 3.4.or 5.)+P(answering 1,2 and also 3,4,5 )

P(answering 1,2 and exactly 2 of 3.4.or 5.)=
P(1,2,3,4 correct, 5 wrong)+P(1,2,3,5 correct, 4 wrong)+P(1,2,4,5 correct, 3 wrong)
also P(1,2,3,4 c, 5w)=P(1,2,3,5 c 4w)=P(1,2,4,5 c 3w )
so  
P(answering 1,2 and exactly 2 of 3.4.or 5.)=3*P(1,2,3,4)=3*1/2*1/2*1/3*1/3*2/3=1/4*2/9=2/36=1/18

note: P(1 correct)=1/2
         P(2 correct)=1/2
         P(3 correct)=1/3
         P(4 correct)=1/3
         P(5 wrong) = 2/3

P(answering 1,2 and also 3,4,5 )=1/2*1/2*1/3*1/3*1/3=1/108

Ans: P= 1/18+1/108=(6+1)/108=7/108