A student takes an exam containing 14 multiple-choice questions. Each question has 4 possible answers. At least 8 correct answers are required to pass.

Question

A student takes an exam containing 14 multiple-choice questions. Each question has 4 possible answers. At least 8 correct answers are required to pass. 

a. If a student guesses, what is the probability that he/she will pass? Round your answers to at least 7 decimal places. 

 

b. What is the expected number of correct answers that will result from this student's strategy? Do not round your answer.

Answer 1

Part A:If we can assume all the 'usual' things: like right answers are random variable, and they are independent and Identically distributed: (i.i.d.), the probability of any of the 4 answers being correct is 1/4 (25%). Since you need to get at 7 answer correctly and it doesn't matter which 7 are guessed correctly, meaning order is not important. For times when order is not important, the way to 'count' the ways you can get at least 'n' of the 14 answers correct by purely guessing, is to use a combination (vs. a permutation). I'll let you look up the formula to calculate a combination of '14 chose x'. Since the problem want to know the probability of passing, we need to calculate the combinations for choosing 8, 9, 10, 11, and all 12 answers correctly. After you've done that, you need to add those 5 probabilities.

Part B:The expected value of a variable is its mean, and the expected value of the sum of random variables

[which you have:(E(8)+E(9)+...+E(12)} is equal to the sum of their individual expected values, You'll need to go through a process similar to what you did for part a.

It's a lot of calculations, but I've told you how to do it, so you should be able to do the rest.