Solutions
180B PRACTICE PROBLEMS FOR FINAL
Please simplify your answers to the extent reasonable without a calculator, show your
work, and explain your answers, concisely.
If you set up an integral or a sum that you
cannot evaluate, leave it as it is; and if the result is needed for the next part, say how you
would use the result if you had it.
1. Suppose Bob is trying to guess a specific natural number
x
*
∈ {
1
, . . ., N
}
.
On his
first guess he chooses a number
X
0
uniformly at random. For
t
∈
N
, if
X
t
=
x
*
the
game ends; if
X
t
negationslash
=
x
*
, he guesses
X
t
+1
uniformly at random from among the numbers
different from
X
t
.
a. [5 points] What is the expected number of guesses it takes Bob to find
x
*
?

b. [5 points] Suppose Bob has a bad memory and can’t remember the number he guessed
previously, so that he guesses
X
t
+1
uniformly at random from among
{
1
, . . ., N
}
. In
this case what is the expected number of guesses it takes him to find
x
*
?

Solutions
180B PRACTICE PROBLEMS FOR FINAL
c. [5 points] Suppose Bob has a good memory, and at each step guesses uniformly at
random among the numbers he has not guessed at any previous step. Now what is
the expected number of guesses it takes him to find
x
*
?

2. [15 points] 2
≤
n
∈
N
is a
prime
number if its only divisors are 1 and itself. The Prime
Number Theorem says that the primes are distributed approximately as if they came
from an inhomogeneous Poisson process
P
(
x
) with intensity
λ
(
x
) = 1
/
ln
x
. [We have
to say approximately since (1) the primes are integers, not general real numbers, and
(2) the primes take determined, not random, values.] Use this theorem to estimate the
number of primes in the interval [2
, N
].