If a student randomly guesses at 7 multiplechoice questions

If a student randomly guesses at 7 multiple-choice questions, find the probability that the student gets exactly 5 or 6 correct. Each question has four possible choices.

Solution

1.

Here, p = 1/4 - 0.25.

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 =    20      
p = the probability of a success =    0.25      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   7   ) =    0.898188143
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.101811857 [ANSWER]

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2.

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.25      
x = the maximum number of successes =    5      
          
Then the cumulative probability is          
          
P(at most   5   ) =    0.617172654 [ANSWER]

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3.

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 =    20      
p = the probability of a success =    0.25      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.202331152 [ANSWER]

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4.

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.25      
x = our critical value of successes =    7      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   7   ) =    0.898188143
          
Thus, the probability of at least   8   successes is  
          
P(more than   7   ) =    0.101811857 [ANSWER]