OR 3500/5500, Summer’08, Homework 1
Homework 1
Due on Tuesday, May 27, 11am.
For each problem just giving the answer will not suﬃce; a proper argument is
required.
Problem 1
A fair die is rolled twice.
(a) List the elements in the following events:
•
A
=At least one of the rolls is 6.
•
B
=The sum of the rolls is 8.
•
C
=Product of the two rolls is divisible by 6.
(b) Compute
P
(
A
),
P
(
B
) and
P
(
C
) using the classical deﬁnition of probability.
Solution
(a)
A
=
{
(1
,
6)
,
(2
,
6)
,
(3
,
6)
,
(4
,
6)
,
(5
,
6)
,
(6
,
1)
,
(6
,
2)
,
(6
,
3)
,
(6
,
4)
,
(6
,
5)
,
(6
,
6)
}
B
=
{
(2
,
6)
,
(3
,
5)
,
(4
,
4)
,
(5
,
3)
,
(6
,
2)
}
C
=
{
(1
,
6)
,
(2
,
3)
,
(2
,
6)
,
(3
,
2)
,
(3
,
4)
,
(3
,
6)
,
(4
,
3)
,
(4
,
6)
,
(5
,
6)
,
(6
,
1)
,
(6
,
2)
,
(6
,
3)
,
(6
,
4)
,
(6
,
5)
,
(6
,
6)
}
(b)Sample space is:
Ω =
{
(
i,j
) : 1
≤
i,j
≤
6
}
Clearly #Ω = 36 and hence
P
(
A
) = 11
/
36
P
(
B
) = 5
/
36
P
(
C
) = 5
/
12
Problem 2
Prove rigorously, using the axioms and derived properties of probability