A multiple choice test contains 10 questions Each questions

A multiple choice test contains 10 questions. Each questions has four choices, but only one of which is correct. A student forgot to study for the test and radnomly guessed an answer for each question.

a. Find the probability that he got no correct answer.

b. Find the probability that in this test he got a grade B or better or equivalently answered 8 or more questions correctly.

Solution

a)

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 =    10      
p = the probability of a success =    0.25      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.056313515 [ANSWER]

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b)

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 =    10      
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   ) =    0.999584198
          
Thus, the probability of at least   8   successes is  
          
P(at least   8   ) =    0.000415802 [ANSWER]