\documentclass[12pt]{article} \usepackage[margin=1.15in]{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 3, Fall 2017}} \bigskip \noindent \textbf{Due}: Friday 9/29 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. 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}. \medskip \noin The following problems are from chapters 2 and 3 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 2.34) Suppose that there are two types of drivers: good drivers and bad drivers. Let $G$ be the event that a certain man is a good driver, $A$ be the event that he gets into a car accident next year, and $B$ be the event that he gets into a car accident the following year. Let $P(G)=g$ and $P(A|G)=P(B|G)=p_1, P(A|G^c)=P(B|G^c)=p_2,$ with $p_1 < p_2$. Suppose that given the information of whether or not the man is a good driver, $A$ and $B$ are independent (for simplicity and to avoid being morbid, assume that the accidents being considered are minor and wouldn't make the man unable to drive). \medskip \noin (a) Explain intuitively whether or not $A$ and $B$ are independent. \medskip \noin (b) Find $P(G|A^c)$. \medskip \noin (c) Find $P(B|A^c)$. \bigskip \noin 2. (BH 2.45) A gambler repeatedly plays a game where in each round, he wins a dollar with probability $1/3$ and loses a dollar with probability $2/3$. His strategy is ``quit when he is ahead by \$2", though some suspect he is a gambling addict anyway. Suppose that he starts with a million dollars. Show that the probability that he'll ever be ahead by \$2 is less than $1/4$. \bigskip \noin 3. (BH 2.55) A certain hereditary disease can be passed from a mother to her children. Given that the mother has the disease, her children independently will have it with probability $1/2$. Given that she doesn't have the disease, her children won't have it either. A certain mother, who has probability $1/3$ of having the disease, has two children. \medskip \noin (a) Find the probability that neither child has the disease. \medskip \noin (b) Is whether the elder child has the disease independent of whether the younger child has the disease? Explain. \medskip \noin (c) The elder child is found not to have the disease. A week later, the younger child is also found not to have the disease. Given this information, find the probability that the mother has the disease. \bigskip \noin 4. (BH 2.65) A standard deck of cards will be shuffled and then the cards will be turned over one at a time until the first ace is revealed. Let $B$ be the event that the \emph{next} card in the deck will also be an ace. \medskip \noin (a) Intuitively, how do you think $P(B)$ compares in size with $1/13$ (the overall proportion of aces in a deck of cards)? Explain your intuition. (Give an intuitive discussion rather than a mathematical calculation; the goal here is to describe your intuition explicitly.) \medskip \noin (b) Let $C_j$ be the event that the first ace is at position $j$ in the deck. Find $P(B|C_j)$ in terms of $j$, fully simplified. \medskip \noin (c) Using the law of total probability, find an expression for $P(B)$ as a sum. (The sum can be left unsimplified, but it should be something that could easily be computed in software such as R that can calculate sums.) \medskip \noin (d) Find a fully simplified expression for $P(B)$ using a symmetry argument. \smallskip \noin Hint: If you were deciding whether to bet on the next card after the first ace being an ace or to bet on the last card in the deck being an ace, would you have a preference? \bigskip \noin 5. (BH 3.1) People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let $X$ be the number of people needed to obtain a birthday match, i.e., before person $X$ arrives there are no two people with the same birthday, but when person $X$ arrives there is a match. Find the PMF of $X$. \bigskip \noin 6. (BH 3.17) An airline overbooks a flight, selling more tickets for the flight than there are seats on the plane (figuring that it's likely that some people won't show up). The plane has 100 seats, and 110 people have booked the flight. Each person will show up for the flight with probability 0.9, independently. Find the probability that there will be enough seats for everyone who shows up for the flight. (Give an exact answer as a sum, and a number obtained by computing the sum using, e.g., R or Wolfram Alpha.) \bigskip \noin 7. (BH 3.20) Suppose that a lottery ticket has probability $p$ of being a winning ticket, independently of other tickets. A gambler buys $3$ tickets, hoping this will triple the chance of having at least one winning ticket. \medskip \noin (a) What is the distribution of how many of the $3$ tickets are winning tickets? \medskip \noin (b) Show that the probability that at least $1$ of the $3$ tickets is winning is $3p-3p^2+p^3$, in two different ways: by using inclusion-exclusion, and by taking the complement of the desired event and then using the PMF of a certain named distribution. \medskip \noin (c) Show that the gambler's chances of having at least one winning ticket do not quite triple (compared with buying only one ticket), but that they do \emph{approximately} triple if $p$ is small. \end{document}