September1999 CMSC 203 / 0201 Fall 2002 Week #9 – 21/23/25 October 2002 Prof. Marie desJardins

September1999 October 1999 TOPICS  Permutations  Combinations  Binomial theorem  Discrete probability  Probability theory

September1999 MON 10/21 PERMUTATIONS AND COMBINATIONS (4.3)

September1999 October 1999 Concepts/Vocabulary  Permutation, r-permutation  P(n, r) = n! / (n-r)!  r-combination  C(n, r) = (n choose r) = n! / (r! (n-r)!)  Pascal’s identity  (n+1 choose k) = (n choose k-1) + (n choose k)  Pascal’s triangle  Binomial theorem  (x+y) n =  j=0 n (n choose j) x n-j y j

September1999 October 1999 Examples  Exercise 4.3.1: List all the permutations of {a,b,c}.  Exercise 4.3.2: How many permutations are there of the set {a,b,c,d,e,f,g}?  How many permutations of a set of size k?  Exercise 4.3.3: How many permutations of {a,b,c,d,e,f,g} end with a?

September1999 October 1999 Examples II  Exercise : A club has 25 members.  (a) How many ways are there to choose four members of the club to serve on an executive committee?  HINT: Which individual is in each of the four positions doesn’t matter  (b) How many ways are there to choose a president, vice president, secretary, and treasurer of the club?  HINT: Which individual is in each of the four positions does matter  Proof of binomial theorem (page 256)

September1999 WED 10/23 DISCRETE PROBABILITY (4.4) ** HOMEWORK #6 DUE ** ** UNGRADED QUIZ TODAY!**

September1999 October 1999 Concepts / Vocabulary  Experiment, sample space, event  Laplace’s probability – p(E) = |E| / |S|  OK for finitely many equally likely outcomes  p(~E) = 1 – P(E)  p(E 1  E 2 ) = p(E 1 ) + p(E 2 ) when E 1, E 2 are disjoint

September1999 October 1999 Examples  Exercise : A roulette wheel has 38 numbers (18 red, 18 black, and 0 and 00, which are neither black or red). The probability that when the wheel is spun it lands on any particular number is 1/38. What is the probability that the wheel …  (a) …lands on a red number?  (b) … lands on a black number twice in a row?  (c) … lands on 0 or 00?  (d) … does not land on 0 or 00 five times in a row?  (e) … lands on a number between 1 and 6, inclusive, on one spin, but does not land between them on the next spin?

September1999 October 1999 Examples II  Example (an apparent paradox): You choose what’s behind Door #1. Before showing you what’s there, the host tells you that Door #2 is a losing door. Should you stick with Door #1 or switch with Door #3?  A door is a door is a door, isn’t it…? Shouldn’t the probability that the prize is behind Door #1 and the probability that the prize is behind Door #3 both be ½, since there are two doors left? (Exercise )  Why does telling you something about Door #2 tell you anything about Door #3??

September1999 October 1999 Examples III  Exercise : Two events E 1 and E 2 are called independent if p(E 1  E 2 ) = p(E 1 ) p(E 2 ). If a coin is tossed three times, which of the following pairs of events are independent:  (a) E 1 : the first coin comes up tails; E 2 : the second coin comes up heads.  (b) E 1 : the first coin comes up tails; E 2 : two, but not three, heads come up in a row.  (c) E 1 : the second coin comes up tails; E 2 : two, and not three, heads come up in a row.

September1999 FRI 10/11 - MON 10/14 PROBABILITY THEORY (4.5)

September1999 October 1999 Concepts and Vocabulary  Axioms of probability: for a set of mutually exclusive outcomes s  S,  0  p(s)  1  s  S p(s) = 1  Event: set of outcomes  Conditional probability p(E|F) = p(E  F) / p(F)  Independence p(E  F) = p(E) p(F), or p(E|F) = p(E)  Bernoulli trials (2 outcomes)  Binomial distribution b(k:n, p) = (n choose k) p k q n-k  Random variables, expected values  Independent random variables, variance

September1999 October 1999 Examples  Dice rolling:  Find the probability of each outcome when a biased die is rolled, if rolling a 2 or 4 are each three times as likely as rolling each of the other four numbers on the die.  What is the probability of rolling a 7 with two ordinary dice?  Exercise 4.5.5: Suppose a pair of dice is loaded. The probability that a 4 appears on the first die is 2/7 (other outcomes are 1/7), and the probability that a 3 appears on the second die is 2/7 (other outcomes are 1/7). What is the probability of rolling a 7 with these two dice?

September1999 October 1999 Examples II  Independence  Exercise : Show that if E and F are independent events, then ~E and ~F are also independent events.  Exercise : If E and F are independent events, prove or disprove that ~E and F are necessarily independent events.  Conditional probability  Exercise : What is the conditional probability that exactly four heads apppear when a fair coin is flipped five times, given that the first flip came up heads?  Exercise : What is the c.p. that a random bit string of length 4 contains at lest 2 consecutive 0s, given that the first bit is a 1?

September1999 October 1999 Examples III  Bernoulli trials: Exercise : Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p.  (a) the probability of no successes  (b) the probability of at least one success  (c) the probability of at most one success  (d) the probability of at least two successes

September1999 October 1999 Examples IV  Random variables and expected values:  Exercise : What is the expected sum of the numbers that appear on two dice, each biased so that a 3 comes up twice as often as each other number?  Exercise : What is the expected value of a $1 lottery ticket when the purchaser wins $10,000,000 iff the ticket contains the six winning numbers chosen from the set {1,2,3,…,50} (and nothing otherwise)?  Exercise : The 203 final exam contains 50 T/F questions (2 points each) and 25 multiple-choice questions (4 points each).  Linda answers a T/F question correctly with probability.9, and a multiple-choice question with probability.8. What is her expected score on the final?  What is the expected score of Emily, who hasn’t studied at all and answers T/F questions correctly with probability.5, and multiple-choice questions correctly with probability.25?