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.

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