\documentclass[12pt]{article} \usepackage[margin=1.23in]{geometry} \geometry{letterpaper} \usepackage{graphicx} \usepackage{amsmath, amssymb, amsthm} \usepackage{hyperref, multicol} \hypersetup{colorlinks=false, allcolors=blue} \newcommand{\noin}{\noindent} \begin{document} \noindent {\large \textbf{Stat 110 Homework 1, Fall 2017}} \bigskip \noindent \textbf{Due}: Friday 9/15 at 5:00 pm, submitted as a PDF via the \href{https://canvas.harvard.edu/courses/27764}{{course webpage}}. Please check carefully to make sure you upload the correct file. Your submission must be a single PDF file, no more than $20$ MB in size. It can be typeset or scanned, but must be clear and easily legible (not blurry or faint) and correctly rotated (e.g., not upside down). No submissions on paper or by email will be accepted. A useful site for compressing and performing various conversions for PDFs is \href{https://smallpdf.com}{smallpdf.com}. If you would like to typeset your work, I recommend using LaTeX. An online LaTeX environment for working on this homework is at \url{https://goo.gl/P52bsA}. If you would like to scan your work, scanners are available in various libraries and computer labs, or you can use a scanner app such as \href{https://evernote.com/products/scannable}{Scannable} (for iPhone) or \nobreak \href{https://www.camscanner.com}{CamScanner} (for Android or iPhone). If you are using a phone to scan, please use a scanner app rather than just taking pictures with the camera. \medskip \noin The following problems are from Chapter 1 of the book. We write BH 1.14, for example, to denote Exercise 14 of Chapter 1 of the book. Please show your work and give clear, careful, convincing justifications (using \emph{words and sentences} to explain your logic, not just formulas). See the syllabus for the collaboration policy. \bigskip \noin 1. (BH 1.2) (a) How many 7-digit phone numbers are possible, assuming that the first digit can't be a $0$ or a $1$? \medskip \noin (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 $911$). \medskip \noin 2. (BH 1.12) Four players, named A, B, C, and D, are playing a card game. A standard, well-shuffled deck of cards is dealt to the players (so each player receives a 13-card hand). \medskip \noin (a) How many possibilities are there for the hand that player A will get? (Within a hand, the order in which cards were received doesn't matter. You can leave your answer in terms of factorials or binomial coefficients.) \medskip \noin (b) How many possibilities are there overall for what hands everyone will get, assuming that it matters which player gets which hand, but not the order of cards within a hand? (Give an answer in terms of factorials or binomial coefficients, and a numerical answer in a form like $4.25 \times 10^{27}$.) \medskip \noin (c) Explain intuitively why the answer to Part (b) is not the fourth power of the answer to Part (a). \medskip \noin 3. (BH 1.14) You are ordering two pizzas. A pizza can be small, medium, large, or extra large, with any combination of 8 possible toppings (getting no toppings is allowed, as is getting all 8). How many possibilities are there for your two pizzas? (Give an answer in terms of binomial coefficients, and the simplified, exact number.) \medskip \noin 4. (BH 1.17) Give a story proof that $$\sum_{k=1}^n k \binom{n}{k} ^2 = n \binom{2n-1}{n-1},$$ for all positive integers $n$. \medskip \noin Hint: Consider choosing a committee of size $n$ from two groups of size $n$ each, where only one of the two groups has people eligible to become president. \medskip \noin 5. (BH 1.35) A deck of cards is shuffled well. The cards are dealt one by one, until the first time an ace appears. \medskip \noin (a) Find the probability that no kings, queens, or jacks appear before the first ace. \medskip \noin (b) Find the probability that exactly one king, exactly one queen, and exactly one jack appear (in any order) before the first ace. \medskip \noin 6. (BH 1.50) A certain class has $20$ students, and meets on Mondays and Wednesdays in a classroom with exactly $20$ seats. In a certain week, everyone in the class attends both days. On both days, the students choose their seats completely randomly (with one student per seat). Find the probability that no one sits in the same seat on both days of that week. (Give an exact answer and a simple approximation in terms of $e$.) \medskip \noin 7. (BH 1.55) Take a deep breath before attempting this problem. In the book \emph{Innumeracy}, John Allen Paulos writes: \begin{quote} Now for better news of a kind of immortal persistence. First, take a deep breath. Assume Shakespeare's account is accurate and Julius Caesar gasped [``Et tu, Brute!"] before breathing his last. What are the chances you just inhaled a molecule which Caesar exhaled in his dying breath? \end{quote} \medskip \noin Assume that one breath of air contains $10^{22}$ molecules, and that there are $10^{44}$ molecules in the atmosphere. (These are slightly simpler numbers than the estimates that Paulos gives; for the purposes of this problem, assume that these are exact. Of course, in reality there are many complications such as different types of molecules in the atmosphere, chemical reactions, variation in lung capacities, etc.) \medskip \noin Suppose that the molecules in the atmosphere now are the same as those in the atmosphere when Caesar was alive, and that in the 2000 years or so since Caesar, these molecules have been scattered completely randomly through the atmosphere. You can also assume that sampling-by-breathing is with replacement (sampling without replacement makes more sense but with replacement is easier to work with, and is a very good approximation since the number of molecules in the atmosphere is so much larger than the number of molecules in one breath). \medskip \noin Find the probability that at least one molecule in the breath you just took was shared with Caesar's last breath, and give a simple approximation in terms of $e$. \end{document}