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Probability and Computing : Solutions to Selected Exercises
Michael Mitzenmacher and Eli Upfal
Do Not Distribute!!!
(Current) List of Solved Exercises
Chapter 1 : 1.1 1.2 1.4 1.5 1.6 1.9 1.12 1.13 1.15 1.18 1.21 1.22 1.23 1.25 1.26
Chapter 2 : 2.1 2.5 2.6 2.7 2.8 2.10 2.13 2.14 2.15 2.17 2.18 2.19 2.20 2.21 2.24 2.32
Chapter 3 : 3.2 3.3 3.7 3.10 3.11 3.12 3.16 3.17 3.20 3.25
Chapter 4 : 4.3 4.6 4.7 4.9 4.10 4.11 4.12 4.13 4.19 4.21
Chapter 5 : 5.1 5.4 5.7 5.9 5.11 5.16 5.21 5.22 5.23 [Discussion of Exploratory Assignment]
Chapter 6 : 6.2 6.3 6.4 6.5 6.9 6.10 6.13 6.14 6.16
Chapter 7 : 7.2 7.12 7.17 7.20 7.21 7.23 7.26 7.27
Weare still in the process of adding solutions. Expect further updates over time.
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Chapter 1
Exercise 1.1: We ip a fair coin ten times. Find the probability of the following events.
(a) The number of heads and the number of tails are equal.
(b) There are more heads than tails.
(c) The ith ip and the (11i)th ip are the same for i =1,...,5.
(d) We ip at least four consecutive heads.
Solution to Exercise 1.1:
(a) The probability is 10 /210 = 252/1024= 63/256.
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(b) By part (a), the probability that the number of heads and the number of tails is different is 193/256.
By symmettry, the probability that there are more heads than tails is 1/2 of this, or 193/512.
(c) The probability that each pair is the same is 1/2; by independence, the probability that all ve pairs
are the same is 1/32.
(d) While there are other ways of solving this problem, with only 1024 possibilities, perhaps the easiest
way is just to exhaustively consider all 1024 possibilities (by computer!). This gives 251/1024.
Exercise 1.2: We roll two standard six-sided dice. Find the probability of the following events, assuming
that the outcomes of the rolls are independent.
(a) The two dice show the same number.
(b) The number that appears on the rst die is larger than the number on the second.
(c) The sum of the dice is even.
(d) The product of the dice is a perfect square.
Solution to Exercise 1.2:
(a) The probability the second die matches the rst is just 1/6.
(b) By part (a), the probability that the rolls are different is 5/6. By symmettry, the probability that the
rst die is larger is 5/12.
(c) The probability is 1/2. This can be done by considering all possibilities exhaustively. Alternatively,
if the rst die comes up with an odd number, the probability is 1/2 that the second die is also odd
and the sum is even. Similarly, if the rst die comes up with an even number, the probability is 1/2
that the second die is also even and the sum is even. Regardless of the outcome of the rst roll, the
probability is 1/2.
(d) The possible squares are 1, 4, 9, 16, 25, and 36. There is 1 way for the product to be 1, 9, 16, 25, or
36, and 3 ways for the product to be 4. This gives a total probability of 8/36 = 2/9.
Exercise 1.4: We are playing a tournament in which we stop as soon as one of us wins n games. We are
evenly matched, so each one of us wins any game with probability 1/2, independently of other games. What
is the probability that the loser has won k games when the match is over?
Solution to Exercise 1.4: Suppose that you win, and I have won k0, nd an upper bound on the probability
that there is a sequence of log n + k consecutive heads.
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Solution to Exercise 1.9: Let us assume n is a power of 2. The probaibility of having log2n+k consecutive
heads starting from the jth ip is
1 = 1 .
log n+k k
2 2 2 n
Here j can range from 1 to nlog2nk+1;forj>nlog2nk+1, there are not enough ips to have
k in a row before reaching the nth ip. Using a union bound, we have an upper bound of
nlog2nk+1≤ 1 .
2kn 2k
Exercise 1.12: The following problem is known as the Monty Hall problem, after the host of the game
show Lets Make a Deal. There are three curtains. Behind one curtain is a new car, and behind the other
two are goats. The game is played as follows. The contestant chooses the curtain that she thinks the car is
behind. Monty then opens one of the other curtains to show a goat. (Monty may have more than one goat
to choose from; in this case, assume he chooses which goat to show uniformly at random.) The contestant
can then stay with the curtain she originally chose or switch to the other unopened curtain. After that, the
location of the car is revealed, and the contestant wins the car or the remaining goat. Should the contestant
switch curtains or not, or does it make no difference?
Solution to Exercise 1.12:
We assume that the car is behing a curtain chosen uniformly at random. Since the contestant can pick
from 1 car and 2 goats, the probability of choosing a curtain with a car behind it is 1/3 and that of choosing
a curtain with a goat behind it is 2/3; if the contestant doesnt switch, the probability of winning is just 1/3.
Now notice that if the contestant switches, she will win whenever she started out by picking a goat! It
follows that switching wins 2/3 of the time, and switching is a much better strategy.
This can be also set up as a conditional probability question, but the above argument is perhaps the
simplest; or simply playing the game several times (using cards
an Ace for the car, deuces for the goats)
can be very convincing.
Exercise 1.13: A medical company touts its new test for a certain genetic disorder. The false negative rate
is small: if you have the disorder, the probability that the test returns a positive result is 0.999. The false
positive rate is also small: if you do not have the disorder, the probability that the test returns a positive
result is only 0.005. Assume that 2 percent of the population has the disorder. If a person chosen uniformly
from the population is tested, and the result comes back positive, what is the probability that the person
has the disorder?
Solution to Exercise 1.13: Let X be the event that the person has the disorder, and Y be the event that
the test result is positive. Then
Pr(X | Y)=Pr(X ∩Y) = Pr(X ∩Y)¯ .
Pr(Y) Pr(X ∩Y)+Pr(X∩Y)
Wenowplug in the appropriate numbers:
Pr(X | Y)= 0.02· 0.999 = 999 ≈0.803.
0.02· 0.999+0.98·0.005 1244
Exercise 1.15: Suppose that we roll ten standard six-sided dice. What is the probability that their sum
will be divisible by 6, assuming that the rolls are independent? (Hint: use the principle of deferred decisions,
and consider the situation after rolling all but one of the dice.)
Solution to Exercise 1.15: The answer is 1/6. Using the principle of deferred decisions, consider the
situation after the rst 9 die rolls. The remainder when dividing the sum of the nine rolls by 6 takes on one
of the values 0,1,2,3,4, or 5. If the remainder is 0, the last roll needs to be a 6 to have the nal sum be
divisible by 6; if the remainder is 1, the last roll needs to be a 5 to have the nal sum be divisible by 6; and
so on. For any value of the remainder, there is exactly one roll which will make the nal sum divisible by 6,
so the probability is 1/6. (If desired, this can be made more formal using conditional probability.)
Exercise 1.18: We have a function F : {0,...,n1}→{0,...,m1}. We know that for 0 ≤ x,y ≤ n1,
F((x+y)modn)=(F(x)+F(y))modm. The only way we have to evaluate F is to use a look-up table
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