Week 10 - Monday

 What did we talk about last time?  Combinations  Binomial theorem

 A bundle of 120 wires has been laid underground between two telephone exchanges 10 miles apart  Unfortunately, it was discovered that the individual wires are not labeled  Visually, there is no way of knowing which wire is which, making connections at either end impossible  Your job is to label the wires at both ends  Walking is your only transportation  You have a battery and a light bulb to test continuity  You have tape and a pen for labeling the wires  What is the shortest distance in miles you will need to walk to correctly identify and label each wire?

 Consider the numbers 1 through 99,999 in their ordinary decimal representations.  How many contain exactly one of each of the digits 2, 3, 4, and 5?  For example, 53,142 counts but 53,541 does not

 On an 8 × 8 chessboard, a rook is allowed to move any number of squares either horizontally or vertically.  How many different paths can a rook follow from the bottom-left square of the board to the top-right square of the board if all moves are to the right or upward?  Hint: Think of representing each move as an R or a U

 A bakery produces six different kinds of pastry, one of which is eclairs. Assume there are at least 20 pastries of each kind.  How many different selections of twenty pastries are there?  How many different selections of twenty pastries are there if at least three must be eclairs?  How many different selections of twenty pastries contain at most two eclairs?

 How many different solutions are there to the following equation, assuming that each x i is a nonnegative integer? x 1 + x 2 + x 3 = 20  What if each x i is a positive integer?

 a + b is called a binomial  Using combinations (or Pascal's Triangle) it is easy to compute (a + b) n

 Compute (2x + 3) 7 using the binomial theorem

Student Lecture

 If n pigeons fly into m pigeonholes, where n > m, then there is at least one pigeonhole with two or more pigeons in it  More formally, if a function has a larger domain than co-domain, it cannot be one-to-one  We cannot say exactly how many pigeons are in any given holes  Some holes may be empty  But, at least one hole will have at least two pigeons

 A sock drawer has white socks, black socks, and red argyle socks, all mixed together,  What is the smallest number of socks you need to pull out to be guaranteed a matching pair?  Let A = {1, 2, 3, 4, 5, 6, 7, 8}  If you select five distinct elements from A, must it be the case that some pair of integers from the five you selected will sum to 9?

 If n pigeons fly into m pigeonholes, and for some positive integer k, n > km, then at least one pigeonhole contains k + 1 or more pigeons in it  Example:  In a group of 85 people, at least 4 must have the same last initial

 Let A and B be events in the sample space S  0 ≤ P(A) ≤ 1  P(  ) = 0 and P(S) = 1  If A  B = , then P(A  B) = P(A) + P(B)  It is clear then that P(A c ) = 1 – P(A)  More generally, P(A  B) = P(A) + P(B) – P(A  B)  All of these axioms can be derived from set theory and the definition of probability

 What is the probability that a card drawn randomly from an Anglo-American 52 card deck is a face card (jack, queen, or king) or is red (hearts or diamonds)?  Hint:  Compute the probability that it is a face card  Compute the probability that it is red  Compute the probability that it is both

 Expected value is one of the most important concepts in probability, especially if you want to gamble  The expected value is simply the sum of all events, weighted by their probabilities  If you have n outcomes with real number values a 1, a 2, a 3, … a n, each of which has probability p 1, p 2, p 3, … p n, then the expected value is:

 A normal American roulette wheel has 38 numbers: 1 through 36, 0, and 00  18 numbers are red, 18 numbers are black, and 0 and 00 are green  The best strategy you can have is always betting on black (or red)  If you bet $1 on black and win, you get $1, but you lose your dollar if it lands red  What is the expected value of a bet?

 Given that some event A has happened, the probability that some event B will happen is called conditional probability  This probability is:

 Given two, fair, 6-sided dice, what is the probability that the sum of the numbers they show when rolled is 8, given that both of the numbers are even?

 Let sample space S be a union of mutually disjoint events B 1, B 2, B 3, … B n  Let A be an event in S  Let A and B 1 through B n have non-zero probabilities  For B k where 1 ≤ k ≤ n

 Bayes' theorem is often used to evaluate tests that can have false positives and false negatives  Consider a test for a disease that 1 in 5000 people have  The false positive rate is 3%  The false negative rate is 1%  What's the probability that a person who tests positive for the disease has the disease?  Let A be the event that the person tests positively for the disease  Let B 1 be the event that the person actually has the disease  Let B 2 be the event that the person does not have the disease  Apply Bayes' theorem

 If events A and B are events in a sample space S, then these events are independent if and only if P(A  B) = P(A)∙P(B)  This should be clear from conditional probability  If A and B are independent, then P(B|A) = P(B)

 Finish probability  Graph basics

 Finish reading Chapter 9  Work on Homework 8  Due next Friday  Start reading Chapter 10