What Is the Probability of Guessing Correctly on a Mixed Multiple Choice Test?

[skhan]

A multiple choice test contains 12 questions, 8 of which have 4 answers each to choose from and 4 of which have 5 answers to choose from. If a student randomly guesses all of his answers, what is the probability that he will get exactly 2 of the 4 answer questions correct and at least 3 of the 5 answer questions correct?
ANS: 0.2552

Heres what I did:

Out of the 8-four answer questions, the student gets 2 of them = (8C2)
Out of the 4-five answer questions, the student gets 3 of them = (4C3)

Therefore:

[8C2(.25)^2(.75)^6][ 4C3(.2)^3(.8)^1+ 4C4(.2)^4(.8)^0].

So:

The probability of getting exactly two of the eight four-option questions is 0.31146240234375.

The probability of getting at least three of the four five-option questions is 0.0272.

Those two are clearly independent. The product of those probabilities is 0.00847177734375.

BUT...the answer is suppose to be 0.2552 apparently.

Any input?

 

[Krusty]

Your method for this question seems right. All that I've found is that maybe the question was not written correctly. The answer given corresponds exactly for the situation where the student guesses exactly two questions from the first set (4 options) and at most 1 from the second set (5 options). This would mean having 2 correct guesses from the first set but not more than 3 correct answers overall. Hope this helped