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[Fall 2012] Exam #1 probability exam

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    Qatar University College of Engineering GENG 200 Probability and Statistics for Engineers Fall 2012: Midterm Exam 1-KeySolution Faculty: Dr. Adel Elomri Date: 07-October-2012 Exercise 1 (30 pts) A bag contains 30 balls 20 of them are Red (R) and 10 are White (W). 2 balls selected randomly without replacement from the bag constitute the sample under study. 1- What are the possible outcomes (determine the sample space S)? S= {(R,W), (R,R), (W,R), (W,W)} 2- Let Y  be the random variable that counts the number of White Balls in the Sample, write down the possible values of   Y. Y=0, 1, 2   3-Deterrmine the Probability Mass Function (    of Y     ,     ,       4- Determine the Cumulative Distribution Function (   of   Y [        GENG 200 Probability and Statistics for Engineers, Midterm Exam 1-fall 2012   Page | 2/4 Exercise 2(30 pts) A dice game consists in rolling a fair six-sided die: the roll is considered as success (win) only when the side number one (face N 1) is on the top, otherwise the die roll is considered as loss (failure). 1-Calculate (with justification) the following: -Probability of a success: 1/6   -Probability of a loss: 5/6    2-Assuming that the die rolls are independent, what is the distribution of the number of trials until getting the first success , (identify the distribution’s parameters)? The number of trials until the first win is following a geometric distribution with parameter P=1/6 3-What is the expected number of trials until a successful die roll? E[X]=1/P=6 4-What is the probability to win at most after 3 trials?        Exercise 3 (30 pts)  Suppose the number of customers that enter a bank in one hour is a Poisson random variable with a mean of 2. 1- How many customers are expected to enter the bank during 6 hours? E[X]=2*6=12   2- What is the probability that at least 3 customers enter the bank during 6 hours?               3- What is the probability that at most one customer enter the bank during 6 hours?             GENG 200 Probability and Statistics for Engineers, Midterm Exam 1-fall 2012   Page | 3/4 4- Determine the length of an interval of time such that the probability that no customers enter the bank during this interval is 0.10. Let I the unknown interval of time:        Exercise 4(15pts) The Qatar Stars League, formerly known as the Q-League, is the highest professional league in Qatari football that currently features 12 clubs. Therefore 6 matches   are weekly  played. Each match (game) has only three possible outcomes: team 1 wins, team 2 wins, or the game is drawn [both teams have scored an equal number of goals].  For a given week, assume someone just guesses (chooses the game outcome randomly) on each of the six matches result, and let X the random variable that counts the number of correct guessed results. The matches are supposed to be played independently. 1-Considering only one match, what is the probability of guessing the correct match’s result?   Guessing on match’s result, means choosing randomly one outcome out of 3 outcomes, so the probability of the correct answer will be 1/3 . P=1/3 2-What are the possible values of X?  X=0, 1, 2, 3,4,5,6 3- What is the distribution of X, identify its parameters?  X is following a binomial distribution with a parameter n=6, and P=1/3 4-What is the expected number of the correct guessed results? E[X]=n P= 6. 1/3=2 5-What is the probability that at most two results are guessed correctly?      ()   ()   ()   ()    
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