The quiz consists of 4 multiplechoice questions Each questio

The quiz consists of 4 multiple-choice questions. Each question has 5 choices for an answer, only one of which is correct. Suppose an unprepared student makes random guesses. Find the probability that the student will get at least two correct answers. You have to define an appropriate random variable, state its distribution and the associated parameter value(s), and express the event of Interest in terms of the random variable.

Solution

Let

x = the number of correct answers

And is binomially distributed with n = 4, p = 1/5 = 0.2.

Now, we want P(x>=2).

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    4      
p = the probability of a success =    0.2      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.8192
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.1808 [ANSWER]