Problem 1. A couple decides to have children until they have both a boy and a girl. What is the expected number of children that they'll end up with? Assume that each child is equally likely to be a boy or a girl and genders are mutually independent Solution. There are many ways to solve

Notes for Recitation 14 Counting Rules Rule 1(Generalized Product Rule). Let be a set of length-k sequences. If there are: n1 possible first entries, n2 possible second entries for each first entry, n3 possible third entries for each combination of first and second entries, etc. then:

Guessing a particular solution. Recall that a general linear recurrence has the form: f(n)=a1f(n-1)+a2f(n-2)+…+aaf(n-d)+g(n) As explained in lecture, one step in solving this recurrence is finding a particular solu- tion; i.e., a function f(n)that satisfies the recurrence, but may not be consistent with the boundary conditions. Here's a recipe to help you guess a particular solution:

Notes for recitation 5 1 Well-ordering principle Every non-empty set of natural numbers has a minimum element Do you believe this statement? Seems obvious, right? Well, it is. But dont fail to realize how tight it is. Crucially, it talks about a non-empty set -otherwise, it would clearly be false. And it also talks about natural

Sums, Approximations, and Asymptotics II Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table's edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+1)-st block;

1 The Number-Picking Game Here is a game that you and I could play that reveals a strange property of expectation. 3, First, you think of a probability density function on the natural numbers. Your distri- bution can be absolutely anything you like. For example, you might choose a uniform distribution on 1, 2, ... 6, like the outcome of a fair die roll. Or you might choose a bi- probability, provided that,...,n. You can even give every natural number a non-zero nomial distribution on 0, 1 he sum of all probabilities is 1

1 Induction Recall the principle of induction: Principle of Induction. Let P(n) be a predicate. If ·P(0) is true,an for all nE N, P(n) implies P(n+1), then P(n) is true for all nE N As an example let's try to find a simple expression equal to the following sum and then use induction to prove our guess correct 1·2+2·3+3:4+…+n·(mn+1) To help find an equivalent expression, we could try evaluating the sum for some small n and(with the help of a computer) some larger n sum

Key to practice exercise I. GRAMMAR 1.C 2.A在 hope,Ibet结构后的that-分句中可使用一般现在时表示将来的动作。如: bet it rains tomorrow. 3.B.主句中若有 require, demand, suggest insist order, prefer, propose recommend, request等动词,其后的that分句中应使用be型虚拟式,即动次原形或 should+动词原形

地球内部过程 Tsunamis海啸 and Coastal Flooding Coastal areas, especially around the Pacific Ocean basin where so many large earthquakes occur, may also be vulnerable易于to tsunamis

Microelectronics Processing Technology Spring Term, 200. Quiz 1 90 min/90 points total March 19 2003 Note: Be brief and relevant in your answers, and use sketches. Show your work. k=1.3810-23J/K=8.6210-eV/K 40 1. Film deposition, CVD )Discuss briefly what factors affect the microstructure a film grown on a substrate