An exam consists of 9 tasks multiple choice questions each

An exam consists of 9 tasks – multiple choice questions, each of the questions lists five answer options. You decided to conduct an experiment and try to GUESS an answer for each of the questions, you do not even read them no to spoil the experiment. Is this a discrete or continuous probability? What subtype is this probability? What is the probability that you answer correctly on exactly 4 questions?

Solution

An exam consists of 9 tasks – multiple choice questions, each of the questions lists 5 answer options.

let X be the random variable denoting the number of correct answers guessed.

so the possible outcomes are

0 answers correctly guessed or 1 answer correctly guessed or 2 answers correctly guessed or,........., or 9 answers correctly guessed.

so X can take either of the values 0,1,2,3,4,5,6,7,8,9

hence it is clear that this is a DISCRETE probability as X can take only some specified distinct values.

the subtype is BINOMIAL PROBABILITY

because here X is nothing but the number of success( correct answers) in 9 independent trials (guessing each question)

so X follows a Binomial distribution with parameters n=9 and p=1/5 [as there are 5 answer options. so the probability of guessing the correct answer out of 5 answers is 1/5]

so X~Bin(9,1/5)

so the mass function is P[X=x]=⁹C_x(1/5)^x(1-1/5)^9-x      x=0,1,2,3,4,5,6,7,8,9

so the probability that you answer correctly on exactly 4 questions is

P[X=4]=⁹C₄(1/5)⁴(1-1/5)^9-4=0.06606 [answer]