1 Independent Events Suppose that we flip two fair coins simultaneously on opposite sides of room. Intu- itively, the way one coin lands does not affect the way the other coin lands. The mathe- matical concept that captures this intuition is called independence. In particular, events A and B are independent if and only if:

Introduction to Probability Probability is the last topic in this course and perhaps the most important. Many Igorithms rely on randomization. Investigating their correctness and performance re- quires probability theory. Moreover, many aspects of computer systems, such as memory

Conditional Probability Suppose that we pick a random person in the world. Everyone has an equal chance of being selected. Let A be the event that the person is an MIT student, and let B be the event that the person lives in Cambridge. What are the probabilities of these events? Intuitively

Generating functions Generating functions are one of the most surprising, useful, and clever inventions in discrete math. Roughly speaking, generating functions transform problems about se- quences into problems about functions. This is great because weve got piles of mathe- matical machinery for manipulating functions. Thanks to generating functions, we can

Counting III Today we'll briefly review some facts you dervied in recitation on Friday and then turn to some applications of counting. 1 The Bookkeeper Rule In recitation you learned that the number of ways to rearrange the letters in the wore BOOKKEEPER is:

We realize everyone has been working pretty hard this term, and were considering Warding some prizes for truly exceptional coursework. Here are some possible categories Best Administrative Critique We asserted that the quiz was closed-book. On the cover

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;

In this example, the domain is the set fa,b, c, d, el and the range is the set Y= (1, 2, 3, 4, 5/. Related elements are joined by an arrow. This relation is a function because every element on the left is related to exactly one element on the right. In graph-theoretic

Sums and Approximations When you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a load-balancing or communications scheme, you'll rarely get a simple answer. The world is not so kind. More likely, you'll end up with a complicated sum:

Recursion-breaking an object down into smaller objects of the same typeis a ma- jor theme in mathematics and computer science. For example, in an induction proof we establish the truth of a statement()from the truth of the statement P(n-1). In pro- gramming, a recursive algorithm solves a problem by applying itself to smaller instances