An exam consists of 15 questions, and each question either comes from a textbook

Question

An exam consists of 15 questions, and each question either comes from a textbook problem or not. Each problem on the exam has a.8 chance of being a textbook problem, and each problem chosen is independent of the other problems. Calculate the probability that the exam consists of exactly 10 textbook problems. Calculate the probability that the exam has more than two textbook problems. How many questions on the exam should one expect to be textbook problems? What is the variance of this random variable and what are the units of variance?

Explanation / Answer

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 =    15      
p = the probability of a success =    0.8      
x = the number of successes =    10      
          
Thus, the probability is          
          
P (    10   ) =    0.103182294 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.8      
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   ) =    5.70491E-08
          
Thus, the probability of at least   3   successes is  
          
P(more than   2   ) =    0.999999943 [ANSWER]

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c)

E(x) = n p = 15*0.8 = 12 [ANSWER]

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d)

Var(x) = n p (1-p) = 15*0.8*(1-0.8) = 2.4 [ANSWER]

It is unitless, as these are just plain numbers.

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