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PROBLEM: Taking Multiple Choice Tests-- Suppose A Student X takes a MCQ with 5 questions on it -- but he has zero knowledge. On every question, he makes a random guess among the four choices. X needs a 80% score to pass-- what is the chance that he will pass the exam?

To answer this question, we must FIRST create a MODEL for the situation. The model is based on certain ASSUMPTIONS which may not hold in the real world. We assume that X makes a simple random choice -- choosing all four options with equal probability. This way, he has exactly 1/4 chance of choosing the right answer. Assuming independence of choices in the MODEL should be a reasonable approximation in this model, and is essential in order to allow use to make the Bernoulli Model applicable, and to make the calculations

AFTER all these model assumptions are in place, we can make the calculations required -- NOTE that we have no guarantee that these will actually apply to the real world situation, which could be different from many reasons.

SOLUTION: After creating a model, we make calculation within the model, which are easy applications of the probability laws.

The chances of getting the first question right are 1/4. This is true for all questions. We can think of the scores of X as a Bernoulli sequence. The outcome 1 of correct answer has 1/4, while the outcome 0 of a wrong answer has 3/4 probability. The chances of ALL five correct answers is the PRODUCT of the separate probabilities

(1/4) x (1/4) x (1/4) x (1/4) x (1/4) = 1/1024

This is the probability of the sequence {1,1,1,1,1}. Next we need to calculate the probability of 1 mistake. There are FIVE sequences which have one mistake: {0,1,1,1,1} {1,0,1,1,1} and so on. Each sequence corresponds to getting a wrong answer on the 1st, 2nd, 3rd 4th or 5th question, and getting all others right. For each sequence, we can calculate the probability by the multiplication rule:

(3/4) x (1/4) x (1/4) x (1/4) x (1/4) = 3/1024, this is the probability of getting the FIRST question wrong, and all others right. A similar calculation shows that each of the five possibilities has 3/1024 probability. Since all five are Mutually Exclusive events, we ADD the probabilities to get 15/1024 as the total probability of exactly one mistake. This means that the probability of 80% or more score is 16/1024 which is between 1% and 2% -- it does not look good.