11/12
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
Assignment 1
Due Date: September 26, 2011
(Hand in your solutions for Questions 3, 4, 13, 16, 18, 32, 37, 40)
1.
A total of 7 different gifts are to be distributed among 13 children. How many distinct results
are possible if no child is to receive more than one gift?
2.
From a group of 8 women and 6 men a committee consisting of 4 men and 4 women is to be
formed. How many different committees are possible if
(a) 2 of the men refuse to serve together;
(b) 2 of the women refuse to serve together;
(c) 1 man and 1 woman refuse to serve together?
3.
In how many ways can 12 people be seated in a row if
(a) there are no restrictions on the seating arrangement;
(b) persons A and B must sit next to each other;
(c) there are 6 men and 6 women and no 2 men or 2 women can sit next to each other;
(d) there are 5 men and they must sit next to each other;
(e) there are 6 married couples and each couple must sit together?
4. Four separate awards (best scholarship, best leadership qualities, and so on) are to be
presented to selected students from a class of 25. How many different outcomes are possible
if
(a) a student can receive any number of awards;
(b) each student can receive at most 1 award?
5. (
Poker
)
If it is assumed that all
poker hands are equally likely, what is the probability of being
dealt
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
5
52
(a) one pair? (This occurs when the cards have denominations
a, a, b, c, d
where
a, b, c,
and
d
are all distinct.)
(b) two pairs? (This occurs when the cards have denominations
a, a, b, b, c
, where
a, b,
and
c
are all distinct.)
(c) three of a kind? (This occurs when the cards have denominations
a, a, a, b, c
.)
(d) a straight? (Five consecutive cards of any suits which are not all the same, including
10, J,
Q, K, A
.)
(e) a flush? (Five cards of the same suit and not all in consecutive order.)
(f) full house? (This occurs when the cards have denominations
a, a, a, b, b
.)
(g) four of a kind? (This occurs when the cards have denominations
a, a, a, a, b
.)
(h)
a straight-flush? (Five consecutive cards of the same suit, including
10, J, Q, K, A
.)
6.
From a group of 4 lawyers, 8 accountants, and 7 doctors, a committee of three is selected at
random. What is the probability that the committee has more lawyers than doctors?
P. 1