1. A True-False test was developed for a Risk Management class. A student, who didn't study, decided to randomly guess the answer on each question. Assume that the probability that the student guess correctly on each question is 50%. The exam has 20 questions. A correct answer adds 1 to the test score, an incorrect answer adds 0.
a) What is the probability that the student has 3, 5, 12, 15, 18 and 20 correct answers?
b) What is the probability that the student has between 5 and 12 correct answers (including 5 and 12)?
What is the probability that the student scores 17 and above ?
c) What is the probability that the student answers correctly more than 70% of the questions ? Compared with the case that the exam has 40 or 80 questions.
d) Formulate similar questions as (a) above for the case of multiple-choice test, in which the answers could be either A, B, C or D
Assume that a True-False test of 20 questions is a sample of size N=20 from a Bernoulli process with p=0.5. Therefore, we can use the Binomial probability distribution p(r|N=20,p=0.5) to evaluate the probability that the student has r correct answers.
a)What is the probability that the student has 3, 5, 12, 15, 18 and 20 correct answers?
p(r=3 |N=20,p=0.5) = 0.0011
p(r=5 |N=20,p=0.5) = 0.0148
p(r=12|N=20,p=0.5) = 0.1201
p(r=15|N=20,p=0.5) = 0.0148
p(r=18|N=20,p=0.5) = 0.0002
p(r=20|N=20,p=0.5) = 0.000001
The following exercise works through several problems that involve calculating probabilities for a Bernoulli distribution.