P ROBLEMS FOR THE MIDDLE TERM EXAM
[1] Given three sets
A, B, C
. Use the Venn diagram to illustrate the following sets:
A
∪
B
∪
C
,
A
∩
B
∪
B
∩
(
C
\
(
A
∪
B
))
.
[2] Let a card be selected from two ordinary
packs of 52 cards. Denote
A
=
{
the card is Diamonds or clubs
}
,
B
=
{
the card is a red
suit one
}
, C
=
{
the card is not a Heart Jack
}
.
Compute the following probabilities:
P
(
A
)
, P
(
B
∪
C
)
, P
(
C

B
) [3]
EX 1( 2 points)
: What are a sample space, events an their probilitiy? Formulate
axioms of probability ? Give an example ?
EX 2 ( 2 points)
: What is the conditional probability of an event
A
given
E
.
When three events
A
i
, i
= 1
,
2
,
3 are independent ? Give an example of a sequence of
dependent ( nonindependent) events
A
i
, i
3 such that each pair of its membrs are
independent.
EX 3 (2 points)
: A pair of fair dice is tossed. We obtain the finite equiprobable
Ω consisting of the 36 ordered pairs of numbers between 1 and 6
Ω =
{
(1
,
1)
,
(1
,
2)
,
· · ·
,
(6
,
6)
}
.
Let
X
assign to each point (
a, b
) in Ω the minimum of its numbers, i.e
X
(
a, b
) =
min(
a, b
). What is the image set of random variable
X
? Find the distribution of
X
and its expectation value, variance and standard deviaton.
EX 3 ( 2 points)
Let a pair of fair dice be tossed.
If the sum is 5, find the
probability that one of the dice is a 2.
EX 4( 2 points)
: Give a definition of a random variabe
X
on a sample space Ω
and its distribution function and, emphasize the above concepts for the case that if Ω
is a discrete space.
EX 5( 2points)
: A drawer contains red socks and white socks. When two socks
are drawn at random, the probability that both socks are red and the third one is white
is
1
2
. (i) How small can the number of socks in the drawer be ? (ii) If the number of
white is divisible by 3, how small can the number of socks be ?
1
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EX 6 (2’)
:
A pair of fair dice is tossed.
We obtain the finite equiprobable Ω
consisting of the 36 ordered pairs of numbers between 1 and 6
Ω =
{
(1
,
1)
,
(1
,
2)
,
· · ·
,
(6
,
6)
}
.
Let
X
assign to each point (
a, b
) in Ω the minimum of its numbers, i.e
X
(
a, b
) =
min(
a, b
). What is the image set of random variable
X
? Find the distribution of
X
and its expectation value, variance and standard deviaton.
EX 7
: (4.20) Box A contains nine cards numbered 1through 9,and bx B contains
five cards numbered 1 through 5. A box is chosen at random and a card drawn. if the
number is even, find the probability that the card came from box A.
EX 8
:(4.23) Let A be the event that a family has children of both sexes, and
let B denote the event that a family has at most one boy. The events A and B are
independent. How small can the number of children in a family be ?
EX 9
Three machines A,B and C produce respectively 50%, 30% and 20% of total
number of items of a factory. The percentages of defective output of these machines are
respectively 2%
,
3%
and
4%. An item is selected at random and is found defective.
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 Spring '09
 thu
 Sets, Normal Distribution, Probability theory

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