A biology quiz consists of 10 multiple-choice questions. 5 must be answered correctly to receive a passing grade. If each question has 4 possible answers, of which only one is correct, what is the probability that a student who guesses at random on each question will pass the examination?
A) 0.9803
B) 0.9965
C) 0781
D) 0.0781
E) 0.0004

Question

A biology quiz consists of 10 multiple-choice questions. 5 must be answered correctly to receive a passing grade. If each question has 4 possible answers, of which only one is correct, what is the probability that a student who guesses at random on each question will pass the examination? ​
A) 0.9803
B) 0.9965
C) 0781
D) 0.0781
E) 0.0004

Quizplus · Accepted Answer

The probability of guessing one question correctly is 1/4. Since there are 10 questions, the probability of guessing all of them correctly is (1/4)^10. 
The probability of guessing 5 or more questions correctly can be found using the binomial distribution: 
P(X ≥ 5) = 1 - P(X < 5) 
= 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)] 
= 1 - [(10C0 * (1/4)^0 * (3/4)^10) + (10C1 * (1/4)^1 * (3/4)^9) + (10C2 * (1/4)^2 * (3/4)^8) + (10C3 * (1/4)^3 * (3/4)^7) + (10C4 * (1/4)^4 * (3/4)^6)] 
= 0.0781

A Biology Quiz Consists of 10 Multiple-Choice Questions