A test consists of 20 multiple choice questions. Each question has 5 possible an

Question

A test consists of 20 multiple choice questions. Each question has 5 possible answers, of which only one is correct. If a student guesses on each question, find the following?

A) The probability that he will guess all of them wrong?

B) the probability that he will guess at most 15 correct?

C) The probability that he will guess at lesst one correct

D) the mean and the standard deviation of the number of correct answers

E) Estimate the probability of the number of correct answers that will fall within the limits ±2

Explanation / Answer

Hence, the probability of a right answer is p = 1/5 = 0.2.

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 =    20      
p = the probability of a success =    0.2      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.011529215 [ANSWER]

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b)

Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    20      
p = the probability of a success =    0.2      
x = the maximum number of successes =    15      
          
Then the cumulative probability is          
          
P(at most   15   ) =    0.999999986 [ANSWER]

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c)

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 =    0.2      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.011529215

Thus, P(at least one) = 1 - P(0) =   0.988470785 [ANSWER]

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d)

Here,

u = mean = np =    4 [ANSWER]
  
s = standard deviation = sqrt(np(1-p)) =    1.788854382 [ANSWER]

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e)

By empirical rule (68-95-99.7 rule), that is around 95%. [ANSWER]

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Hi! For part e), if you use another method to approximate this in class, please resubmit this question indicating what method you use in class. That way we can continue helping you! Thanks!

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