Problem 1Problem Set 9
Problem 2
Let n be a positive
integer. What is the largest binomial
coefficient C(n,r),
where r is a nonnegative integer less than or
equal to n?
Prove your answer is correct.
Problem 3
Give a formula for the coefficient
of x^k in the expansion
of (x + 1/x)^100, where
k is an integer.
Problem 4
The following procedure is
used to break ties in games in the
championship round of the
World Cup soccer tournament. Each team
selects five players.
These two groups of 5 alternate
taking penalty kicks (one
per player). If the score is still tied at the end of the ten
penalty kicks, this procedure
is repeated. If the score is still tied
after 20 penalty kicks,
a sudden-death shootout occurs, with the first
team scoring an unanswered
goal victorious.
A scoring scenario
consists of a given order of making / not making the
penalty kicks. (for
example, first player scores and all the rest of the
players miss)
a)
How many different scoring scenarios are possible if the game is
settled in the first round
of 10 penalty kicks? (the round ends
once it is impossible for
a team to equal the number of goals scored
by the other team)
b)
How many different scoring scenarios for the first and second
groups of penalty kicks
are possible if the game is settled in the
second round of 10 penalty
kicks?
c)
How many scoring scenarios are possible for the full set of penalty
kicks if the game is settled
with no more than 10 total additional
kicks after the two rounds
of 5 kicks for each team?
Problem 5
February 29 occurs only in
leap years. Years divisible by 4, but
not by 100, are always leap
year. Years divisible by 100, but not by
400, are not leap years,
but years divisible by 400 are leap years.
a) What probablility distribution for birthdays should be used to
reflect how often February
29 occurs?
b)
Using this probability distribution, what is the probablity that in
a group of n people, there
at least two with the same birthday?
Problem 6
Suppose that 0.1% of the
population has a rare blood disease. There is
a test for this blood disease,
that is correct 95% of the time. Given that
a particular patient has
tested positive, what are his actual odds of having
the disease?
Problem 7
In how many ways can a dozen
books be placed on four
distinguishable shelves
a) if the books are indistinguishable copies of the same title?
b)
if no two books are the same, and the positions of the books on the
shelves matter?
Problem 8
a)
Let n and r be positive integers. Explain why the number of
solutions of the equation
x1 + x2 + ... + xn = r, where xi is a
nonnegative integer for
i = 1, 2, 3, ..., n, equals the number of
r-combinations of a set
with n elements.
b)
How many solutions in nonnegative integers are there to the
equation x1 + x2 + x3 +
x4 = 17?
c)
How many solutions in positive integers are there to the equation
in part (b)?
Problem 9
Problem 10
Show that in any set of n
+ 1 positive integers not exceeding 2n
there must be two that are
relatively prime.
Problem 11
A space probe near Neptune
communicates with Earth using bit
strings. Suppose that
in its transmissions it sends a 1 one-third of
the time and a 0 two-thirds
of the time. When a 0 is sent, the
probablity it is received
correctly is 0.9, and the probability it is
received incorrectly (as
a 1) is 0.1. When a 1 is sent, the
probability it is received
correctly is 0.8, and the probability it is
received incorrectly (as
a 0) is 0.2.
a) What is the probability a 0 is received?
b)
What is the probability a 0 was transmitted, given that a 0 was
received?