Find the probability

Question

A student takes a 15-question, true/false exam and guesses on each question. Find the probability of
passing if the lowest passing grade is 12 correct out of 15. Would you consider this event likely to
occur? Explain your answer.

Answer 1

I think this is a valid way to do this problem, but I'm not 100% sure. Since it hasn't been answered yet, I figured I'd give you my thoughts:

 

 

Since there are 2 possible answers for each of the 15 questions, there are 2¹⁵ = 32768 ways to answer all the questions on the quiz.

 

Let's figure out how many ways there are to answer at least 12 questions correctly. To pass, the student can answer three questions however he wants, but must answer the other 12 correctly. So, he has 2 answer choices for 3 questions, but only 1 answer choice for the other 12. First, let's find out how many ways there are to choose those 12 correct answers:

 

"15 choose 12" = ₁₅C₁₂ = 15!/(12!×3!) = (15×14×13)/(6) = 455 combinations of 12 correct answers

 

 

Now, for any combination of correct answers, the other three questions on the quiz can be answered either correctly or incorrectly. That's a total of 2³ = 8 ways to answer the other three questions.

 

455×8 = 3640 ways to answer at least 12 questions on the quiz correctly.

 

 

That means there are 3640 ways to pass the quiz out of a total of 32768 ways to answer the questions, which leaves the student with a 3640/32768 = 0.11 = 11% chance to pass the test. This isn't likely to occur; it only happens around 1 in 10 times.