A student has taken an exam consisting 10 multiple choice questions. Each question has 4 answer choices and has one correct answer. Questions are independent of each other. If the student answer questions randomly, what is the probability that the student marks less than 3 correct choices?
p correct = .25
1-p = .75
p(0 correct)= C(10,0).25^0*.75^10
p(1 correct)= C(10,1).25^1 *.75^9
p(2 correct)= C(10,2).25^2 *.75^8
c(10,0) = 1 of course
C(10,1) = 10
C(10,2) = 45
add the three probabilities
C(10,1)
To find the probability that the student marks less than 3 correct choices, we need to find the probability of the student marking 0, 1, or 2 correct choices.
Let's calculate the probability for each case separately.
Case 1: Marking 0 correct choices
The probability of marking a correct choice for each question is 1/4, and the probability of marking an incorrect choice is 3/4. Since there are 10 questions and they are independent, the probability of marking 0 correct choices is (3/4) * (3/4) * ... * (3/4) (10 times).
So, the probability of marking 0 correct choices is (3/4)^10.
Case 2: Marking 1 correct choice
The student can choose any 1 question out of the 10 to mark as correct. The probability of marking a correct choice for the chosen question is 1/4, and the probability of marking an incorrect choice for the remaining 9 questions is 3/4. We can use combinations to determine the number of ways to choose 1 question out of 10.
So, the probability of marking 1 correct choice is (10 choose 1) * (1/4) * (3/4)^9.
Case 3: Marking 2 correct choices
The student can choose any 2 questions out of the 10 to mark as correct. The probability of marking a correct choice for the chosen questions is (1/4)^2, and the probability of marking an incorrect choice for the remaining 8 questions is (3/4)^8. We can use combinations to determine the number of ways to choose 2 questions out of 10.
So, the probability of marking 2 correct choices is (10 choose 2) * (1/4)^2 * (3/4)^8.
Now, we just need to add up the probabilities for each case:
Probability of marking less than 3 correct choices = (3/4)^10 + (10 choose 1) * (1/4) * (3/4)^9 + (10 choose 2) * (1/4)^2 * (3/4)^8.