Description
problem 1. A new promotion at Insomnia Cookies lets you play the following game: by paying a fixed
amount of money, you can get a free cookie every day, selected uniformly at random from chocolate
chunk, double chocolate mint, and peanut butter chip, until you get a chocolate chunk (after which the
promotion will stop). Suppose you decide to play this game.
(a) Let X be the number of free cookies you obtain with this promotion. Find Ex(X).
(b) Let Y be the number of free cookies you get that are not chocolate chunk. Find Ex(Y).
Solution:
problem 2. Suppose that the price of a stock starts at $10. Each day, the price of the stock increases by
$1 with probability 1/3 and decreases by $1 with probability 2/3, independently of other days. Find the
probability that after six days the stock has its original price of $10.
Solution:
problem 3. Charles claims that he can distinguish between beer and ale 75 percent of the time. Ruth bets
that he cannot and, in fact, just guesses. To settle this, a bet is made: Charles is to be given ten small
glasses, each having been filled with beer or ale, chosen by tossing a fair coin. He wins the bet if he gets
seven or more correct. Find the probability that Charles wins if he has the ability that he claims. Find the
probability that Ruth wins if Charles is guessing.
Solution:
problem 4 (Problem 19.19 in the textbook). BU students sometimes delay doing laundry until they finish
their problem sets. Assume all random values described below are mutually independent.
(a) A busy student must complete 3 problem sets before doing laundry. Each problem set requires 1 day
with probability 2/3 and 2 days with probability 1/3. Let B be the number of days a busy student
delays laundry. What is Ex(B)?
Example: If the first problem set requires 1 day and the second and third problem sets each require
2 days, then the student delays for B = 5 days.
(b) A relaxed student rolls a fair, 6-sided die in the morning. If he rolls a 1, then he does his laundry
immediately (with zero days of delay). Otherwise, he delays for one day and repeats the experiment
the following morning. Let R be the number of days a relaxed student delays laundry. What is
Ex(R)?
Example: If the student rolls a 2 the first morning, a 5 the second morning, and a 1 the third morning,
then he delays for R = 2 days.
(c) Before doing laundry, an unlucky student must recover from illness. The number of days the student
is ill is determined as follows. We toss a fair coin and we roll a fair 6-sided die. If the coin is heads,
the student is ill for n days, where n is the outcome of the die roll. If the coin is tails, the student is
ill for 2n days, where n is the outcome of the die roll. What is Ex(U)?
Example: If the die roll is 5, the student delays for U = 5 days if the coin toss is heads and U = 10
days if the coin toss is tails.
(d) A student is busy with probability 1/2, relaxed with probability 1/3, and unlucky with probability
1/6. Let D be the number of days the student delays laundry. What is Ex(D)?
Solution:
problem 5. The company RandomCereal places figurines in cereal boxes via the following process: in
each box, with probability 1/2 it places no figurine in the box, and with probability 1/2 it places some
figurine in the box. When it places a figurine, it chooses one of 4 possible types uniformly and random.
(a) Suppose you have collected k of the 4 types of figurines, where k ∈ {0, 1, 2, 3} (collecting a type of
figurine means that you have at least one figurine of that type). What is the expected number of
cereal boxes you need to open until you obtain a new type of figurine (that you have not already
collected)? Your answer should be in terms of k.
(b) Suppose you do not have any of the figurines. What is the expected number of boxes you need to
open to collect all 4 types of figurines?
Solution:
problem 6 (Simulations). Download the HW10 Jupyter notebook. Complete all the exercises in the notebook. Submit the Jupyter notebook with your solutions to the Homework 10 Programming assignment on
Gradescope. Your submission should be a single .ipynb file.
H10-2
