Download First Exam - Electricals 4 You

The Hashemite University
Electrical Engineering Department
Spring 11-12
A
Probability: First Exam َ◌(25%)
Instructor: Dr. A Al-Nimrat.
Student Name:…………..
Serial number: …………..
Table (1)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
[30 points] Q1)Fill-in Table (1) above with the alphabet of the most correct answer for the following
questions:
1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. the probability
that the ticket drawn has a number which is a multiple of 3 or 5 is
a)
1
2
b)
8
15
c)
2
5
d)
9
20
e)None
2.A bag contains 2 red, 3 green, and 2 blue balls. Two balls are drawn at random. the
probability that none of the balls drawn is blue is
10
11
2
5
b)
c)
d)
e) None
a)
21
21
7
7
3. The probability of getting a sum 9 from two throws of a dice is
1
1
1
1
a)
b)
c)
d)
e) None
6
8
9
12
4.In a box, there are 8 red, 7 blue, and 6 green balls. One ball is picked up randomly. the
probability that it is neither red nor green
1
3
8
7
b)
c)
d)
e) None
3
4
21
19
5.Three unbiased coins are tossed. the probability of getting at most two heads is
1
3
7
3
b)
c)
d)
e) None
a)
4
4
8
8
6.In a class, there are 15 boys and 10 girls. Three students are selected at random. The
probability that 1 girl and 2 boys are selected, is
21
1
25
3
a)
b)
c)
d)
e) None
46
50
117
25
a)
1
7. Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl with replacement. the probability of
selecting 2 green marbles and 2 blue marbles is
a) 0.135
b) 0.35
c) 0.432
d) 0.25
e) None
8.From a pack of 52 cards, two cards are drawn together at random. the probability of both the
cards being kings is
1
25
35
1
a)
b)
c)
d)
e) None
15
57
256
221
9. The average number of homes sold by a company is 2 homes per day. the probability that
exactly 3 homes will be sold in 3 days is
a) 0.32
b) 0.026
c) 0.089
d) 0.45
e)None
10. Suppose scores on a test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, then the probability that a person who takes the test will score
between 90 and 110 (hint: F(1)= 0.84) is
b) 0.68
c) 0.45
d) 0.78
e) None
a) 0.52
11. If the probability that a student is absent in class is (0.02), then the probability that there
will be zero absence in a class of 40 students is
a) 1
b) 0.98
c)0.225
d)0.445
e)None
k (1 + x ) : 0 < x < 2
for f x (x ) = 
, answer (12,13)
: o.w
0
12. the value of K that makes f X (x ) a valid pdf equals
a) -0.25
b) 0.25
c) 0.5
d) 1.5
e)None
4
13. the E [X ] equals
•
64
32
45
5
b)
c)
d)
e) None
15
21
15
7
14.The probability that a student is accepted to a college is 0.3. If 5 students from the same
school apply, the probability that at most 2 are accepted is
a) 0.246
b) 0.580
c)0.837
d)0.345
e)None
a)
15. if X is N(0,1) and Y = X 2 then mean value of Y equals
a) 0
b) 1
c)0.5
d)-1
e)None
16. if the sample space of a random experiment is S = {0,1, 2.5,6} , if the random variable
X (s ) = cos(π s ) , the range of X is
a){-1,0,1}
b){-1,0}
c){-1,1}
d){0,1}
e) None
[10 points] Q2) a random current is described by the sample space S = {−4 ≤ i ≤ 12} , if the
random varable X is defined as
2
 −2 : i ≤ −2
 i : −2 p i ≤ 1

X (i ) = 
1:1 p i ≤ 4
6 : i f 4
Find:
a) The range of R.V X.
b) What is the type of X.
c) The cdf FX ( x )
d) The pdf f X ( x )
Q3) the pdf of X is given by
1 − x2
f X ( x ) = e u ( x ) , if event A =
{1 p X ≤ 3}, B =
{X ≤ 2.5},C =A I B .
2
Find:
a) The cpdf f X ( x / B ) .
b) The probability of C.
3
1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What
is the probability that the ticket drawn has a number which is a multiple of 3 or 5?
1
2
A.
B.
2
5
8
9
C.
D.
15
20
2.A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random.
What is the probability that none of the balls drawn is blue?
10
11
2. A.
B.
21
21
2
5
C.
D.
7
7
3.In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up
randomly. What is the probability that it is neither red nor green?
1
3
A.
B.
3
4
3.
7
8
C.
D.
19
21
9
E.
21
4.What is the probability of getting a sum 9 from two throws of a dice?
1
1
A.
B.
4.
6
8
1
1
C.
D.
9
12
5.Three unbiased coins are tossed. What is the probability of getting at most two
heads?
3
1
5. A.
B.
4
4
3
7
C.
D.
8
8
6.Two dice are thrown simultaneously. What is the probability of getting two
numbers whose product is even?
1
3
A.
B.
2
4
3
5
C.
D.
8
16
7.In a class, there are 15 boys and 10 girls. Three students are selected at random.
The probability that 1 girl and 2 boys are selected, is:
21
25
A.
B.
46
117
1
3
C.
D.
50
25
8.In a lottery, there are 10 prizes and 25 blanks. A lottery is drawn at random.
What is the probability of getting a prize?
1
2
A.
B.
10
5
2
5
C.
D.
7
7
9. From a pack of 52 cards, two cards are drawn together at random. What is the
.
probability of both the cards being kings?
1
25
A.
B.
15
57
35
1
C.
D.
256
221
9.Two dice are tossed. The probability that the total score is a prime number is:
1
5
A.
B.
6
12
1
7
C.
D.
2
9
10.A card is drawn from a pack of 52 cards. The probability of getting a queen of
club or a king of heart is:
1
2
A.
B.
13
13
1
1
C.
D.
26
52
11.A bag contains 4 white, 5 red and 6 blue balls. Three balls are drawn at random
from the bag. The probability that all of them are red, is:
1
3
A.
B.
22
22
2
2
C.
D.
91
77
12.Two cards are drawn together from a pack of 52 cards. The probability that one
is a spade and one is a heart, is:
3
29
A.
B.
20
34
47
13
C.
D.
100
102
13.One card is drawn at random from a pack of 52 cards. What is the probability
that the card drawn is a face card (Jack, Queen and King only)?
1
3
A.
B.
13
13
1
9
C.
D.
4
52
15.A bag contains 6 black and 8 white balls. One ball is drawn at random. What is
the probability that the ball drawn is white?
3
4
A.
B.
4
7
1
3
C.
D.
8
7
Example 2
The probability that a student is accepted to a prestigeous college is 0.3. If 5 students from
the same school apply, what is the probability that at most 2 are accepted?
Solution: To solve this problem, we compute 3 individual probabilities, using the binomial
formula. The sum of all these probabilities is the answer we seek. Thus,
b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3)
b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087
b(x < 2; 5, 0.3) = 0.8369
Suppose we have a bowl with 10 marbles - 2 red marbles, 3 green marbles, and 5 blue
marbles. We randomly select 4 marbles from the bowl, with replacement. What is the
probability of selecting 2 green marbles and 2 blue marbles? P = 0.135
The average number of homes sold by the Acme Realty company is 2 homes per day. What
is the probability that exactly 3 homes will be sold tomorrow? P=0.180
Suppose scores on an IQ test are normally distributed. If the test has a mean of 100 and a
standard deviation of 10, what is the probability that a person who takes the test will score
between 90 and 110?F(1)= 0.84, P( 90 < X < 110 ) = 0.68
y
−
1
X is N(0,1) , Y = X , then f Y ( y ) =
e 2
2π y
2
X is uniform in (-1,2), Y = X 2 , then pdf of y equals: