Homework Problems
Tuesday April 3rd
1. How many ways are there to permute the letters in the word MISSISSIPPI?
2. (a) How many 7-digit phone numbers are possible, assuming that the first digit can’t be a
0 or a 1?
(b) Re-solve (a), except now assume also that the phone number is not allowed to start
with 911 (since this is reserved for emergency use, and it would not be desirable for the system
to wait to see whether more digits were going to be dialed after someone has dialed 911).
5. A knock-out tournament is being held with 2^n tennis players. This means that for each
round, the winners move on to the next round and the losers are eliminated, until only one
person remains. For example, if initially there are 2^4 = 16 players, then there are 8 games in
the first round, then the 8 winners move on to round 2, then the 4 winners move on to round 3,
then the 2 winners move on to round 4, the winner of which is declared the winner of the
tournament. (There are various systems for determining who plays whom within a round, but
these do not matter for this problem.)
(a) How many rounds are there?
(b) Count how many games in total are played, by adding up the numbers of games played in
each round.
(c) Count how many games in total are played, this time by directly thinking about it without
doing almost any calculation. Hint: How many players need to be eliminated?
8. (s) (a) How many ways are there to split a dozen people into 3 teams, where one team has 2
people, and the other two teams have 5 people each?
(b) How many ways are there to split a dozen people into 3 teams, where each team has 4
people?
21. Three people get into an empty elevator at the first floor of a building that has 10 floors.
Each presses the button for their desired floor (unless one of the others has already pressed
that button). Assume that they are equally likely to want to go to floors 2 through 10
(independently of each other). What is the probability that the buttons for 3 consecutive floors
are pressed?
22, (s) A certain family has 6 children, consisting of 3 boys and 3 girls. Assuming that all birth
orders are equally likely, what is the probability that the 3 eldest children are the 3 girls?
27. (s) For each part, decide whether the blank should be filled in with =, <, or >, and give a
clear explanation.
(a) (probability that the total after rolling 4 fair dice is 21) ____ (probability that the total after
rolling 4 fair dice is 22)
(b) (probability that a random 2-letter word is a palindrome1 ) ____ (probability that a random 3-
letter word is a palindrome)
29. (s) Elk dwell in a certain forest. There are N elk, of which a simple random sample of size n
are captured and tagged (“simple random sample” means that all N n sets of n elk are equally
likely). The captured elk are returned to the population, and then a new sample is drawn, this
time with size m. This is an important method that is widely used in ecology, known as capturerecapture. What is the probability that exactly k of the m elk in the new sample were previously
tagged? (Assume that an elk that was captured before doesn’t become more or less likely to be
captured again.)
33. A random 13-card hand is dealt from a standard deck of cards. What is the probability that
the hand contains at least 3 cards of every suit?
42. Let A and B be events. The difference B - A is defined to be the set of all
elements of B that are not in A. Show that if A ⊆ B, then P(B - A) = P(B) - P(A)
43. Let A and B be events. The symmetric difference AΔB is defined to be the set of all elements
that are in A or B but not both. In logic and engineering, this event is also called the XOR
(exclusive or ) of A and B. Show that P(AΔB) = P(A) + P(B) - 2P(A ∩ B), directly using the
axioms of probability.
Tuesday April 10th
45. Events A and B are independent if P(A ∩ B) = P(A)P(B) (independence is
explored in detail in the next chapter).
(a) Give an example of independent events A and B in a finite sample space S (with
neither equal to ∅ or S), and illustrate it with a Pebble World diagram.
(b) Consider the experiment of picking a random point in the rectangle
R = {(x, y) : 0 < x < 1, 0 < y < 1},
where the probability of the point being in any particular region contained within R is the area of
that region. Let A1 and B1 be rectangles contained within R, with areas not equal to 0 or 1. Let A
be the event that the random point is in A1, and B be the event that the random point is in B1.
Give a geometric description of when it is true that A and B are independent. Also, give an
example where they are independent and another example where they are not independent.
48. (s) A card player is dealt a 13-card hand from a well-shuffled, standard deck of cards. What
is the probability that the hand is void in at least one suit (“void in a suit” means having no cards
of that suit)?
37. 51. Fred needs to choose a password for a certain website. Assume that he will choose an
8-character password, and that the legal characters are the lowercase letters a, b, c, . . . , z, the
uppercase letters A, B, C, . . . , Z, and the numbers 0, 1, . . . 9
(a) How many possibilities are there if he is required to have at least one lowercase letter in his
password?
(b) How many possibilities are there if he is required to have at least one lowercase letter and at
least one uppercase letter in his password?
(c) How many possibilities are there if he is required to have at least one lowercase letter, at
least one uppercase letter, and at least one number in his password?
Thursday April 14th
Chapter 2:
1. (s) A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that
80% of email is spam. In 10% of the spam emails, the phrase “free money” is used, whereas
this phrase is only used in 1% of non-spam emails. A new email has just arrived, which does
mention “free money”. What is the probability that it is spam?
2. (s) A woman is pregnant with twin boys. Twins may be either identical or fraternal
(nonidentical). In general, 1/3 of twins born are identical. Obviously, identical twins must be of
the same sex; fraternal twins may or may not be. Assume that identical twins are equally likely
to be both boys or both girls, while for fraternal twins all possibilities are equally likely. Given the
above information, what is the probability that the woman’s twins are identical?
5. Three cards are dealt from a standard, well-shuffled deck. The first two cards are flipped over,
revealing the Ace of Spades as the first card and the 8 of Clubs as the second card. Given this
information, find the probability that the third card is an ace in two ways: using the definition of
conditional probability, and by symmetry
6. A hat contains 100 coins, where 99 are fair but one is double-headed (always landing
Heads). A coin is chosen uniformly at random. The chosen coin is flipped 7 times, and it lands
Heads all 7 times. Given this information, what is the probability that the chosen coin is doubleheaded? (Of course, another approach here would be to look at both sides of the coin—but this
is a metaphorical coin