A multiplechoice test contains 25 questions each with four a

A multiple-choice test contains 25 questions, each with four answers. Assume a student just guesses on each question.

1) What is the probability that the student answers 5 questions correctly?

2) What is the probability that the student answers more than 2 questions correctly?

Solution

The probability of one correct answer is 1/4 or 0.25.

1.

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    25      
p = the probability of a success =    0.25      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.164537588 [answer]

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2.

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    25      
p = the probability of a success =    0.25      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   2   ) =    0.032108521
          
Thus, the probability of at least   3   successes is  
          
P(more than   2   ) =    0.967891479 [answer]