A teacher gives an 11-question true-false pop quiz. What is the probability, as a percentage, that a student randomly answers at least 6 of the questions correctly?

Answer 1

We're going to use the principle of symmetry for this problem! First notice that since it is a true false pop quiz, there is an equal probability of getting a question right or wrong.

Thus, the probability that a student randomly answers 6 questions right has to be the same as the probability of a student randomly answering 6 questions wrong! And remember, answering 6 questions wrong is the same as answering 5 questions right.

Using this principle, we see that answering 7 questions right has the same probability as answering 4 questions right, Pr[8 right] = P[3 right], ..., and Pr[11 right]=Pr[0 right].

Adding up Pr[Answering 6 or more right] = Pr[6]+Pr[7]+...Pr[11] = Pr[0]+Pr[1]+...+Pr[5] = x (our answer)

Since we notice Pr[0]+..+Pr[11] must = 1, or 100%, 2x = 1 and x = 50%

Answer 2

P(X ≥ 6) = ∑¹¹_x=6 11! / [x! (11-x)!] (0.5)^x(0.5)^11-x

Use the binomial probability to determine the chance of getting at least 6 correct answers on a true-false quiz.

The most efficient way will be to use P(X ≤ 5) based on the complement rule.

P(X ≥ 6) = 1 - P(X ≤ 5) = 0.7256

There is a 72.56% chance that a student will get at least 6 correct answers on a true-false quiz.