0 1. For 1 5 i, j 5 3, by (i, j) we mean that Vanns card number is i, and Pauls card number is
j.C1ear1y,A = {(1, 2), (1, 3), (2. 3)} and B = {(2.1),(3,1),(3,2)}.
(2) Since A n B = 0, the events A and B are mutually exclusive.
(b) None of (1, 1), (2, 2),
. 7 .
o 11 . The coefcient of (2203 (4y)4 in the expansion of (2x 4y)7 )5 (4). Thus the coefment
7
of x3y2 in this expansion is 23(-4)4 (4) z 71, 680.
(50) (150)
5 45
o _ W = 0.00206.
16 200)
50
o 22. The answer is the solution of the equation = 20. This
Mot 30530, (9% 2063/ AOMW/IL Ss/Men;
fag/L r?
i I 1 . Let G be the event that Susan is gp
that Robert will commit perjury IS
ilty. Let L be the event that Robert will lie. The probability
6 5 5 4 4 3 3 2 2 1 1
5.a-x._._
U 11 10X9X8X7X6.X5X4X3X2X10.00216
0 3. The set of possible values of X is {0, 1, 2. . . , N). Assuming that people have the disease
independent of each other,
(1~p)p 15i<N
P(X=i)={
(lwp)N i=0.
' 7. Note that X is neither continuous nor discrete. The answers are
(a) F(6) = 1 implies that
Pays M72.-
0 1 . On average, in the long run, the two businesses have the same prot. The one that has a prot
with lower standard deviation should be chosen by Mr. Jones because hes interested in steady
income. Therefore, he should choose the rst business
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Papa 94/3
e330
0' = 1 ~ (2'3 = 0.9502.
CD. A = (0.05)(60) = 3; the answer is I
(5) We call a room success if it is vacant next Saturday; we call it failure if it is occupied.
Assuming that next Saturday is a
Fag i
20'
(D Z (20)(0.25)"(0.75)20-i = 0.0009.
i
i=12
@N (t), the number of customers arriving at the post ofce at or prior to t is a Poisson process
with A = 1/3. Thus
6 6 .(1/3)3o "
P(N(30) 5 6) = Z P(N(30) = i) = Z w = 0.130141.
i=0 i=0 '
160 200
=
Math 30530: Introduction to Probability, Fall 2011
Midterm Exam 1
Solutions
1. For any three events A, B and C, say whether each of the statements below are always true or
sometimes false/sometimes true. If true, give a proof; if sometimes false/sometimes
Introduction to Probability, Fall 2009
Math 30530
Review questions for exam 1 solutions
1. Let A, B and C be events. Some of the following statements are always true, and some are not. For
those that are always true, give a proof. For those that are somet
MATH 550530 (FM 2w? w Hap/twat 09 54mm;
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@ (a) 1/(1/12) = 12. (b) m 0.07.
k 7 3 5 ~ a t 6 w
[5) (2)(0.2) (0.8) ~0.055. ( ) 42 a .
§. We have
@The probability that a random bridge hand has three aces is
4 48
_ 3 10
p 52
13
Therefore, the aver