Steve and Jess are two students who have agreed to take part in a psychology experiment. Each has to answer several sets of multiple-choice questions. Each set has the same number of questions, , where is a number greater than 20. For each question there are four possible options A, B, C or D, of which only one is correct.

1. Steve decides to guess the answer to every question, so that for each question he chooses A, B, C or D at random.
   

    

   Let the random variable be the number of questions that Steve answers correctly in a particular set.

   1. What is the probability that Steve will answer the first three questions of this set correctly?  (1 mark)
      

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   2. Use the fact that the variance of is to show that the value of is 25.  (1 mark)
      

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2. The probability that Jess will answer any question correctly, independently of her answer to any other question, is  . Let the random variable be the number of questions that Jess answers correctly in any set of 25.

   If   , show that the value of  .  (2 marks)
   

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Let    be a discrete random variable with binomial distribution  . The mean and the standard deviation of this distribution are equal.

Given that  , what is the smallest number of trials, , such that  .   (2 marks)

The binomial random variable,  , has    and  

Calculate  .   (3 marks)

Let    be a discrete random variable with a binomial distribution. The mean of    is 1.2 and the variance of    is 0.72

The values of (the number of independent trials) and (the probability of success in each trial) are

A.  

B.  

C.  

D.