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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
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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
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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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