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الواجب
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Question - 1: Consider a multiple choice assessment paper, on which each question has four possible answers. In each case, just one of these answers is correct. A student who is taking this assessment decides to answer the questions in order but to guess the answers at random.

(a) Find the probability that the first correct answer given, is for question 5.


(b) What is the probability that at least one of the questions 1─ 9, is answered correctly?





(b) Approximately how many guesses, on average would have to be made in order to obtain a correct answer? If there are 16 questions in total on the paper, what is the probability of giving incorrect answers to all of them?






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Question - 2: Out of 50 students in an AOU branch, 20 are studying MST121 course, 13 are studying MU120 course, 23 are studying T171 course, 8 studying MST121 and MU120, 12 are studying MST121 and T171, and 8 are studying MU120 and T171, 6 are studying the three courses. If a student is chosen at random,

(a)Find the probability that the student is studying only the MST121 course.








(b)Find the probability that the student is studying, MST121 or T171.










(c) Find the probability that the student is not studying any one of the three courses.













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Question - 3: Consider the functions, , for x in

(a) Show that f(x) is a probability density function in .









(b) Consider the continuous random variable X, with probability density function f(x). Find, P (X= π/9), P(X in ), and P(X in ).










(c) Write the integral formula for the value of the mean, , of the continuous random variable X, with probability density function f(x).
[ Obtaining the value of the mean can be done by using Math Cad].













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Question - 4: A sample of 100 Printer Cartridges of brand A (single use), showed a mean life time efficiency of 42000 print pages with a standard deviation of 1200 print pages, while a sample, of 100 Printer Cartridges of a re-filled type brand B ( after multiple refill), showed a mean life time efficiency of 41500 print pages, with a standard deviation of 1700 print pages.

(a) Use the data above to investigate whether there is a difference between the life time efficiency means of Printer Cartridges Populations of brands A and B(single use vs. multiple use). Establish the null and alternative hypotheses.







(b) Find the two sample Test Statistic Z and decide whether to accept or reject the null hypothesis at the 5% significance level.








(c) Find a, 90%, confidence intervals, for the Printer Cartridges population life efficiency means for both brands, A & B. Ignoring the inconvenience of multiple refill, which brand would you recommend for the university, or a large publishing firm to use?.










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Question - 5: In an experiment, a pair of fair dices with faces numbered, 1 through 6, are tossed, and the absolute difference , , is recorded at each toss.

(a) Find the size of the sample space S (the set of different values of Z), and the median of S. Find the values of Z with the highest frequency (that is find the mode(s) of the sample space S).









(b) List all outcomes in the event A = {Z is an odd number}. Find the probability, P (A). Consider the events B = {Z is a square number}, and C = {Z is a prime number}. Find P (B), P ( C), and P(C B).









(b) Find the expected value, and the variance, for the difference, Z, in this experiment.