Raj is taking a true-false quiz with six questions. The first five questions are each worth 1 point, and the last question is worth 2 points. For any given question, the probability that Raj will answer it correctly is 2/3 and the probability that he'll answer it incorrectly is 1/3 What is the probability that Raj will score exactly 4 points on the quiz?
There are two ways for Raj to score exactly 4 points:
He gets 3 of the first 5 questions correct and the last question incorrect.
He gets 2 of the first 5 questions correct and the last question correct.
The probability of the first case is:
(3/3)^5 * (1/3) = 243/729
The probability of the second case is:
(3/3)^4 * (2/3) = 81/729
The total probability of Raj scoring exactly 4 points is the sum of these two probabilities:
243/729 + 81/729 = 324/729 = 4/9
