In a multiple choice examination there are 20 question, each question has four alternative answer following it and a student must select one correct answer. Four marks are given for the correct answer and one mark is deducted for every wrong answer. A student must score at least 50% of maximum possible marks to pass the examination. Suppose a student has not study at all so he decide to answer the question on the random bases. What is the probability that he we pass the examination?

Question

In a multiple choice examination there are 20 question, each question has four alternative answer following it and a student must select one correct answer. Four marks are given for the correct answer and one mark is deducted for every wrong answer. A student must score at least 50% of maximum possible marks to pass the examination. Suppose a student has not study at all so he decide to answer the question on the random bases. What is the probability that he we pass the examination?

Answer 1

To find the probability that the student will pass the examination by answering questions randomly, we need to determine the minimum number of correct answers required to achieve a passing score.

In this case, the student needs to score at least 50% of the maximum possible marks, which is (20 questions * 4 marks) = 80 marks.

To pass, the student must therefore answer at least (80 / 4 =) 20 questions correctly.

The probability of getting a question right by randomly guessing is 1 out of 4, since there are 4 alternative answers for each question.

Now, let's calculate the probability:

The probability of getting exactly 20 questions correct is (1/4)^20, since the student needs to get all 20 questions correct by guesswork.

However, the student can also get more than 20 questions correct by guesswork, resulting in a higher score. To pass, the student needs to score at least 50% of the maximum marks, so they can answer any number of additional questions correctly as long as they have at least 20 correct answers.

To calculate the probability of getting more than 20 questions correct, we need to sum the probabilities of getting exactly 21, 22, 23, and so on, up to all 20 questions correct.

Therefore, the probability that the student will pass the examination by guessing randomly is:

P(Pass) = P(20 questions correct) + P(21 questions correct) + P(22 questions correct) + ... + P(all 40 questions correct)

P(Pass) = (1/4)^20 + (1/4)^21 + (1/4)^22 + ... + (1/4)^40

Calculating this probability requires evaluating a geometric series. However, it is important to note that the probability decreases exponentially with each additional question, making it highly unlikely for the student to pass the examination by guessing randomly.

Therefore, the actual probability that the student will pass under these conditions is extremely low.